EXHIBIT 10.4

                         REGISTRATION RIGHTS AGREEMENT

       REGISTRATION RIGHTS AGREEMENT dated as of October 26, 1998, between
ALLIANCE RESOURCES PLC, a public limited company duly organized and validly
existing under the laws of England and Wales ("Alliance"), and LASALLE STREET
NATURAL RESOURCES CORPORATION, a Delaware corporation (the "Investor").

       Alliance and the Investor are parties to a Warrant Agreement of even date
herewith (as modified and supplemented and in effect from time to time, the
"Warrant Agreement"), providing for the issuance by Alliance of Warrants (as
hereinafter defined) which entitle the Investor to purchase from Alliance
3,275,000 Ordinary Shares (as defined in the Warrants) on October 30, 1998 (the
"Date of Issuance") as provided in the Warrant Agreement and the Warrants.  In
that connection, Alliance wishes to afford the Investor certain registration
rights in respect of the Ordinary Shares issued or issuable upon exercise of the
Warrants.  Accordingly, the parties hereto agree as follows:

SECTION 1.  Definitions.

       Each capitalized term used herein without definition shall have the
meaning ascribed thereto in the Warrant Agreement.  As used in this Agreement
the following terms have the following meanings:

       "Alliance" shall have the meaning set forth in the preamble of this
Agreement.

       "Commission" means the U.S. Securities and Exchange Commission (or any
successor or similar governmental agency or authority) administering the
Securities Act and/or the Exchange Act.

       "Cutback Registration" means any registration in which the Managing
Underwriter advises Alliance, and Alliance in turn notifies the holders of
Registrable Securities requested to be included therein in accordance with
Section 5.02, that 

 
marketing factors require a limitation of the number of shares of Ordinary
Shares to be underwritten in such registration.

       "Difco Holders" means F. Fox Benton, Jr., Lizinka M. Benton, F. Fox
Benton III, Lizinka C. Benton and Lucia T. Benton, in their capacities as
"Selling Shareholders" as that term is defined in that certain Registration
Rights Agreement, dated as of the 30th day of October, 1998, by and among the
preceding parties and Alliance.

       "Effective Period" has the meaning set forth in Section 5.01(b).

       "Electing Holders" means any Difco Holders or EnCap Holders who have
requested inclusion of shares of Ordinary Shares held by such holder in a
registration.

       "EnCap Holders" means EnCap Investments L.C., EnCap Equity 1996 Limited
Partnership and Energy Capital Investment Company PLC, in their capacities as
"New Shareholders" as that term is defined in that certain Registration Rights
Agreement, dated as of the 30th day of October, 1998, by and among the preceding
parties and Alliance.

       "Exchange Act" means the U.S. Securities Exchange Act of 1934, as
amended, or any similar Federal statute, and the rules and regulations of the
Commission thereunder, all as the same shall be in effect at the time.

       "Indemnified Party" has the meaning set forth in Section 6.03.

       "Indemnifying Party" has the meaning set forth in Section 6.03.

       "Investor" shall have the meaning set forth in the preamble of this
Agreement.

       "Long-Form Requested Registration" shall mean any Requested Registration
that is a registration on a Form S-1 under the Securities Act or any successor
form or similar long-form registration.

       "Managing Underwriter" means, with respect to any registration, the
underwriter or underwriters managing such registration.

       "NASDAQ" has the meaning set forth in Section 12.

       "Ordinary Shares" has the meaning set forth in the Warrant Agreement.

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       "Original Requesting Holder" has the meaning set forth in Section
3.01(a).

       "Other Requesting Holder" has the meaning set forth in Section 3.01(a).

       "Person" means any individual, corporation, association, joint venture,
limited liability company, partnership, trust, business or other entity or
organization, and shall include any government or political subdivision, or any
agency or instrumentality thereof.

       "Piggyback Registration" means any registration which is not a Requested
Registration, other than (a) any registration on a Form S-8 under the Securities
Act (or any successor thereto); (b) any registration relating to an offering to
be made solely to employees (including management or employee incentive plans);
(c) any registration relating to a an offering of securities made by Alliance
solely (except in respect of fractional shares in exchange for securities of
Alliance other than Ordinary Shares including, without limitation, notes or
other debt instruments of Alliance); or (d) any registration on a Form S-4 (or
any successor thereto or other comparable form).

       "Public Offering" means any offering of Ordinary Shares to the public,
either on behalf of Alliance or any of its Stockholders, pursuant to an
effective registration statement under the Securities Act.

       "register," "registered" and "registration" refer to a registration of
Ordinary Shares or other securities of Alliance effected by preparing and filing
a registration statement in compliance with the Securities Act and the
declaration or ordering of the effectiveness of such registration statement.

       "Registrable Securities" means (i) shares of Ordinary Shares issued or
issuable upon exercise of any Warrants, including without limitation any
Ordinary Shares into which such Ordinary Shares may thereafter be changed or
converted, and (ii) any additional shares of Ordinary Shares or other securities
issued or distributed by way of a dividend, stock split or other distribution in
respect of the Ordinary Shares referred to in clause (i) above, or acquired by
way of any rights offering or similar offering made in respect of the Ordinary
Shares referred to in clause (i) above; provided, however, that, as to any such
shares of Ordinary Shares so issued or issuable, such shares will cease to be
Registrable Securities when such shares have been sold to the public pursuant to
a registration or pursuant to Rule 144.

       "Registration Agreements" means the Agreements described in Exhibit A
hereto.

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       "Requested Registration" means a registration requested by holders of
Registrable Securities pursuant to Section 3.

       "Requesting Holder" means any of the original Requesting Holder and the
other Requesting Holders.

       "Rule 144" means Rule 144 promulgated by the Commission under the
Securities Act (or any successor or similar rule then in force).

       "Rule 144A" means Rule 144A promulgated by the Commission under the
Securities Act (or any successor or similar rule then in force).

       "Securities Act" means the U.S. Securities Act of 1933, as amended, or
any similar Federal statute, and the rules and regulations of the Commission
thereunder, all as the same shall be in effect at the time.

       "Short-Form Registration" means a registration of the Ordinary Shares of
Alliance on Form S-2 or Form S-3 under the Securities Act or any successor form
or similar short-form registration.

       "Stockholder" has the meaning set forth in the Warrant Agreement.

       "Warrant Agreement" has the meaning set forth in the preamble of this
Agreement.

       "Warrants" has the meaning set forth in the Warrant Agreement.

       "Withdrawing Holder" has the meaning set forth in Section 3.02.

SECTION 2.  Piggyback Registration.

       If at any time or from time to time after the date hereof Alliance
proposes to effect a Piggyback Registration for its account or for the account
of a security holder or holders (other than holders of Registrable Securities),
then Alliance shall:

       (a) promptly give to each holder of Registrable Securities notice thereof
(which notice shall include a list of the jurisdictions in which Alliance
intends to attempt to qualify such securities under or otherwise comply with the
applicable blue sky or other state securities laws); and

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       (b) include in such Piggyback Registration (and any related qualification
under or other compliance with blue sky or other state securities laws), and in
any underwriting involved therein, all the Registrable Securities specified in a
request, made within 15 days after receipt of such notice from Alliance, by any
holder of Registrable Securities; provided, however, that Alliance shall not be
required to include any securities of holders of Registrable Securities in such
registration unless such holders accept the terms of the underwriting as agreed
upon between Alliance and the underwriters selected by it; and provided,
further, that if such Piggyback Registration is a Cutback Registration, then the
number of shares of Ordinary Shares to be included in the underwriting or
registration shall be allocated first to Alliance, the holders of Registrable
Securities and the Electing Holders (pro rata, based on the total number of
shares of securities of Alliance, including Registrable Securities, requested by
Alliance and each such holder to be included therein); and thereafter to any
other holders requesting inclusion in the registration on the basis of the
number of shares each other requesting holder requests be included bears to the
total number of shares of all other holders of Ordinary Shares that have been
requested be included in such registration. If a person who has requested
inclusion in such registration as provided above does not agree to the terms of
any such underwriting, such person shall be excluded therefrom by written notice
from Alliance, the underwriter, or the holders of Registrable Securities.  The
securities so excluded shall also be withdrawn from registration.

       (c) (i) If, at any time after giving written notice of its intention to
register any of its Ordinary Shares and before the effective date of the
registration statement filed in connection with the registration, Alliance
determines for any reason not to register its Ordinary Shares, Alliance may, at
its election, give written notice of its determination to the holders of
Registrable Securities and the Electing Holders and, thereupon, shall be
relieved of its obligation to register any Registrable Securities in connection
with that registration, without prejudice, however, to the future rights of the
holders of Registrable Securities under this Section, (ii) if Alliance
determines in its discretion to delay the registration of its Ordinary Shares,
Alliance shall be permitted to delay the registration of any Registrable
Securities for the same period as the delay in registering any other Ordinary
Shares, and (iii) Alliance is not required to effect any registration for a
requesting holder of Registrable Securities pursuant to this Section 2 unless it
receives reasonable assurances that the requesting holder of Registrable
Securities will pay any expenses required to be paid by it as a provided in
Section 5.

       (d) The rights of holders with respect to Piggyback Registrations shall
be pari passu with the piggyback registration rights of Difco Holders and the
EnCap Holders.

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SECTION 3.  Requested Registration.

       3.01 Request for Registration.

       (a)  If after the date Alliance shall receive a request from any holder 
of Registrable Securities (including Warrants) that Alliance effect any
registration under the Securities Act to which such holder is entitled under
this Section 3 (including without limitation any related qualification under or
compliance with blue sky or other state securities laws) with respect to all or
a part of the Registrable Securities owned by such holder, then Alliance shall
promptly give notice of such request to each other holder of Registrable
Securities, and Alliance shall thereupon promptly use its best efforts
diligently to effect such Requested Registration and related qualifications and
compliances within 120 days after receiving such request for registration
(including without limitation the execution of an undertaking to file post-
effective amendments and appropriate qualifications under or other compliance
with the applicable blue sky or other state securities laws) as may be
reasonably requested by the holder of Registrable Securities who made the
original request (the "Original Requesting Holder") and by the holders of
Registrable Securities who make requests to Alliance within 15 days after the
giving of the aforesaid notice by Alliance (each of the foregoing an "Other
Requesting Holder") and as would permit or facilitate the sale and distribution
of all or such portion of the Registrable Securities as are specified in any
such request; provided, however, that Alliance shall not be obligated to take
any action to effect a Requested Registration or any related qualification or
compliance pursuant to this Section 3:

            (i)  if the Requesting Holders do not request to include in such
       registration Registrable Securities (issued or issuable on exercise of
       the Warrants) having an aggregate Current Adjustment Price (as defined in
       the Warrant), determined as of the date of the notice from the Original
       Requesting Holder under Section 3.01(a) of (A) at least $750,000 for the
       holders' first Requested Registration or (B) at least $250,000 for the
       holders' second Requested Registration;

            (ii) if Alliance shall have already effected two Requested
       Registrations on behalf of the holders of Registrable Securities pursuant
       to this Section 3.01, each of which Requested Registrations (A) has been
       declared or ordered effective (including without limitation qualification
       under or other compliance with state blue sky or securities laws
       requested) and which effectiveness has not been suspended or stopped by
       any governmental or judicial authority, and (B) remains continuously
       effective for a period of time not less than the Effective Period; or

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            (iii) if, within 30 days after receipt of the initial request of
       the Original Requesting Holder pursuant to this Section 3.01, Alliance
       shall elect to include in such registration Ordinary Shares for its own
       account, whereupon Alliance shall notify each Requesting Holder that
       Alliance has elected to effect a Piggyback Registration and shall
       thereafter diligently proceed to do so, including therein the Registrable
       Securities as to which notice was given by the Requesting Holders
       pursuant to this Section 3.01, but subject to the limitations set forth
       in Section 2(b)(i) (it being understood, however, that such registration
       shall not be deemed to be a Requested Registration for the purposes of
       Sections 3.01(a)(ii) or 3.01(c)).

            (iv)  Notwithstanding the foregoing, (A) Alliance shall not be
       obligated to effect a registration pursuant to this Section 3 during the
       period starting with the date 60 days prior to Alliance's good faith
       estimated date of filing of, and ending on a date 120 days following the
       effective date of, a registration statement pertaining to an underwritten
       public offering of securities for the account of Alliance, provided that
       Alliance is at all times during such period diligently pursuing such
       registration, (B) Alliance shall not be obligated to effect a
       registration of Registrable Securities pursuant to this Section 3
       pursuant to any request of Holders of Registrable Securities if such
       request is received after the receipt by Alliance of a request for
       registration pursuant to one of the Registration Agreements, and any such
       registration pursuant to this Section 3 would likely result in a
       registration statement being declared effective prior to the date that is
       90 days after the effective date of any such registration effected
       pursuant to the Registration Agreement, and (C) Alliance shall not be
       obligated to effect a registration pursuant to this Section 3 and shall
       have the right to defer such filing for a period of not more than 120
       days after receipt of the request of holders of Registrable Securities,
       if Alliance shall furnish to such holders a certificate signed by the
       President of Alliance stating that in the good faith judgment of the
       Board of Directors of Alliance, it would be seriously detrimental to
       Alliance and its shareholders for such registration statement to be filed
       and it is therefore essential to defer the filing of such registration
       statement; provided, however, that, subject to the limitation set forth
       in the proviso in Section 3.01(a)(ii), if Alliance shall no longer be
       eligible to effect a Short-Form Requested Registration following the
       deferral of registration pursuant to this paragraph, then the holders of
       Registrable Securities shall, subject to Section 3.01(a)(ii), be entitled
       to a Long-Form Requested Registration for each such deferral.

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            (v) If at any time after the holders' initial Requested 
       Registration, a request of the holders of Registrable Securities for a
       Requested Registration shall be denied by Alliance solely because the
       aggregate Current Adjustment Price of such Registrable Securities sought
       to be included in such registration is below the requisite dollar amount
       specified in Section 3.01(a)(i)(B), then at the election of the holders
       of a majority of the then outstanding Registrable Securities, and in
       exchange for the right of the holders to request a second Requested
       Registration under this Section 3, the holders shall be entitled to
       convert their Registrable Securities represented by Warrants (having an
       aggregate Current Adjustment Price for all holders of not more than
       $250,000) pursuant to Section 12 of the Warrant.

       (b) If a Requested Registration becomes a Cutback Registration and the
number of shares of Registrable Securities actually sold in such Requested
Registration is not at least a majority of the number of shares of Registrable
Securities requested to be included in such registration, then (A) such
Requested Registration shall not be deemed to be a Requested Registration for
the purposes of Section 3.01(a)(ii); and (B) notwithstanding that such Requested
Registration is a Cutback Registration, Alliance shall continue to use its best
efforts diligently to comply with all its obligations (including without
limitation payment of expenses) under this Agreement with respect to such
Requested Registration.  The registration statement filed pursuant to the
request of holders of Registrable Securities may, subject to the provisions of
Section 3.01(c), include other shares of Ordinary Shares of Alliance, which are
held by persons who, by virtue of agreements with Alliance, are entitled to
include their securities in any such registration, and Alliance shall have the
right to include shares of Ordinary Shares in such registration for its own
account as provided therein.

       (c) If a Requested Registration becomes a Cutback Registration, the
number of Ordinary Shares to be included in the underwriting or registration
shall be allocated first to the holders of Registrable Securities and the
Electing Holders (pro rata, based on the number of Registrable Securities
requested by each such holder to be included therein), second to Alliance and
thereafter to any other holders requesting inclusion in the registration on the
basis of the number of shares each other requesting holder requests be included
bears to the total number of shares of all other holders of Ordinary Shares that
have been requested be included in such registration.  If a person who has
requested inclusion in such registration as provided above does not agree to the
terms of any such underwriting, such person shall be excluded therefrom by
written notice from Alliance, the underwriter, or the holders 

                                      -8-

 
of Registrable Securities. The securities so excluded shall also be withdrawn
from registration.

       3.02   Underwriting.  If Requesting Holders intend to distribute the
Registrable Securities covered by such request by means of an underwriting, the
Requesting Holders shall so advise Alliance as a part of the request made
pursuant to this Section 3, and in such event, the Requesting Holders shall
negotiate in good faith with an underwriter or underwriters proposed by Alliance
to act as the Managing Underwriter in connection with the underwriting of the
Requested Registration; provided, however, that if those Requesting Holders who
hold at least a majority of the Registrable Securities to be included in such
Requested Registration have not agreed with such underwriter or underwriters as
to the terms and conditions of such underwriting within 20 days following
commencement of such negotiations, then the Requesting Holders may select an
underwriter or underwriters of their choice to be the Managing Underwriter,
which choice shall be subject to the approval of the Board of Directors of
Alliance (such approval not to be unreasonably withheld or delayed, taking into
account Alliance's agreements with underwriters then in effect).  Alliance and
the Requesting Holders shall enter into an underwriting agreement in customary
form with the underwriter or underwriters selected for such underwriting (it
being understood that (i) all expenses customarily paid for by the issuer of
securities pursuant to such an underwriting agreement shall be paid for by
Alliance, and (ii) all indemnification obligations which are customarily those
of the issuer of securities under such underwriting agreement shall be the
obligations of Alliance).  If a Requesting Holder disapproves of the terms of an
underwriting (the "Withdrawing Holder"), the Withdrawing Holder may elect to
withdraw therefrom by notice to Alliance and the Managing underwriter; and each
of the remaining Requesting Holders shall be entitled to increase the number of
shares of Registrable Securities being registered to the extent of the shares
withdrawn by the Withdrawing Holder in the proportion which the number of shares
of Registrable Securities being registered by such remaining Requesting Holder
bears to the total number of shares being registered by all such remaining
Requesting Holders; provided, however, that the requirements contained in
Section 3.01(a)(i) shall then be met and subject to Section 3.01(c).

SECTION 4.  Expenses of Registration.

       Except as otherwise provided herein, (a) in the case of Requested
Registrations pursuant to Section 3, all expenses incurred by Alliance or the
Holders in connection with any registration, qualification or compliance
effected pursuant to this Agreement, including without limitation all
registration, filing and qualification fees, printing expenses, fees and
disbursements of counsel for Alliance and for the 

                                      -9-

 
holders of Registrable Securities, and the expenses of any audits required by
such registration, shall be borne by Alliance and (b) in the case of Piggyback
Registrations pursuant to Section 2 (other than a primary registration by
Alliance), all of the incremental expenses incurred by Alliance (and not
otherwise reimbursed) shall be borne by the holders of Registrable Securities
included in any registration pursuant to the terms hereof; provided, however,
that Alliance shall not be required to pay: (i) the underwriters' fees,
discounts or commissions relating to Registrable Securities; or (ii) the fees
and disbursements of more than a single law firm for all holders of Registrable
Securities to be selected by the holders of a majority of such Registrable
Securities participating in a Requested Registration pursuant to Section 3.
Notwithstanding the foregoing, Alliance shall not be required to pay for any
expenses of any registration proceeding begun pursuant to Section 3 if the
request for registration is subsequently withdrawn at the request of the holders
of a majority of the Registrable Securities to be registered therein (which
holders shall bear such expenses), unless the holders of a majority of the
Registrable Securities agree that such registration shall be deemed to
constitute a Requested Registration for purposes of the limitation set forth in
the proviso of Section 3.01(a)(ii); provided, however, that if (a) between the
date such request for registration is made and the date of such withdrawal,
there has occurred a material adverse change in the condition, business or
prospects of Alliance (or a material adverse change occurring prior to such
request is first publicly disclosed) and (b) such withdrawal shall have occurred
prior to the effective date of the applicable registration statement (it being
understood that the holders shall only have the right to withdraw a Requested
Registration prior to such effective date), then the holders shall not be
required to pay any of such expenses and no Requested Registration shall be
deemed to have occurred pursuant to Section 3.

SECTION 5.  Registration Procedures.

       5.01   In the case of each registration, qualification or compliance
effected by Alliance pursuant to this Agreement, Alliance shall, by notice to
each holder of Registrable Securities included in such registration, keep such
holder advised in writing as to the initiation, progress and effective date of
each registration, qualification and compliance, and, at the expense of
Alliance, Alliance will:

       (a) prepare and file with the Commission a registration statement with
respect to such Registrable Securities and use its best efforts to cause such
registration statement to become effective as soon as reasonably practicable
thereafter; and before filing a registration statement or prospectus or any
amendments or supplements thereto, furnish to the Investor (provided Registrable
Securities held by the Investor are covered by such registration statement) and
the 

                                      -10-

 
underwriter or underwriters, if any, copies of all such documents proposed to be
filed, including without limitation documents incorporated by reference in the
prospectus and, if requested by such holders of Registrable Securities, the
exhibits incorporated by reference, and such holders shall have the opportunity
to object to any information pertaining to such holders that is contained
therein and Alliance will make the corrections reasonably requested by an
underwriter or such holders with respect to such information prior to filing any
registration statement or amendment thereto or any prospectus or any supplement
thereto;

       (b) prepare and file with the Commission such amendments and supplements
to such registration statement and the prospectus used in connection therewith
as may be necessary to keep such registration statement effective for a period
(the "Effective Period") of not less than 180 days, or such shorter period as is
necessary to complete the distribution of the securities covered by such
registration statement and comply with the provisions of the Securities Act with
respect to the disposition of all securities covered by such registration
statement during such period in accordance with the intended methods of
disposition by the seller thereof set forth in such registration statement;

       (c) furnish to each seller of Registrable Securities such number of
copies of such registration statement, each amendment and supplement thereto,
the prospectus included in such registration statement (including without
limitation each preliminary prospectus) and such other documents as such seller
may reasonably request in order to facilitate the disposition of the Registrable
Securities owned by such seller;

       (d) use its best efforts to register or qualify such Registrable
Securities under such other securities or blue sky laws of such jurisdictions as
any seller reasonably requests and take such other steps which may be necessary
or advisable in the reasonable judgment of the managing underwriter (and at the
reasonable request of such managing underwriter) to enable such seller to
consummate the disposition in such jurisdictions of the Registrable Securities
owned by such seller (provided that Alliance will not for any such purpose be
required to (1) qualify generally to do business as a foreign corporation in any
jurisdiction where it would not otherwise be required to qualify but for the
requirements of this subsection; (2) subject itself to taxation in any such
jurisdiction; (3) consent to general service of process in any such
jurisdiction; or (4) register or qualify Registrable Securities or take any
other action under the state securities or "Blue Sky" laws of any jurisdiction
if, in the judgment of the Board of Directors of Alliance, the consequences of
the registration, qualification or other action would be unduly burdensome to
Alliance).

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       (e) notify the Investor and each seller of such Registrable Securities
when a prospectus relating thereto is required to be delivered under the
Securities Act, of the occurrence of any event as a result of which the
prospectus included in such registration statement contains an untrue statement
of a material fact or omits any fact necessary to make the statements therein
not misleading, and, at the request of any such seller, or at the request of any
seller upon the happening of any event of the kind described in Section 5.01(k),
Alliance shall prepare a supplement or amendment to such prospectus so that, as
thereafter delivered to the purchasers of such Registrable Securities, such
prospectus shall not contain an untrue statement of a material fact or omit to
state any fact necessary to make the statements therein not misleading or cures
the event of the kind described in Section 5.01(k);

       (f) in the case of an underwritten offering, cause to be delivered to the
sellers of Registrable Securities and the underwriters, if any, opinions of
counsel to Alliance in customary form, covering such matters as are customarily
covered by opinions for an underwritten public offering as the underwriters may
reasonably request and addressed to the underwriters and such sellers;

       (g) make available for inspection by any seller of Registrable Securities
that is a Significant Holder, any underwriter participating in any disposition
pursuant to such registration statement, and any attorney, accountant or other
agent retained by any seller or underwriter, all financial and other records,
pertinent corporate documents and properties of Alliance, and cause Alliance's
officers, directors, employees and independent accountants to supply all
information reasonably requested by any such seller, underwriter, attorney,
accountant or agent in connection with such registration statement;

       (h) provide a transfer agent and registrar for all such Registrable
Securities not later than the effective date of such registration statement;

       (i) cause to be delivered, immediately prior to the effectiveness of the
registration statement (and, in the case of an underwritten offering, at the
time of delivery of any Registrable Securities sold pursuant thereto), letters
from Alliance's independent certified public accountants addressed to each
seller that is a Significant Holder and each underwriter, if any, stating that
such accountants are independent public accountants within the meaning of the
Securities Act and the applicable published rules and regulations thereunder,
and otherwise in customary form and covering such financial and accounting
matters as are customarily covered by letters of the independent certified
public accountants delivered in connection with primary or secondary
underwritten public offerings, as the case may be;

                                      -12-

 
       (j) make generally available to the holders of Registrable Securities a
consolidated earnings statement (which need not be audited) for the 12 months
beginning after the effective date of a registration statement as soon as
reasonably practicable after the end of such period, which earnings statement
shall satisfy Section 11(a) of the Securities Act and Rule 158 thereunder; and

       (k) promptly notify the Investor and any Significant Holder selling
Registrable Securities in such registration and the underwriter or underwriters,
if any:

              (i)   when the registration statement, any preeffective amendment,
       the prospectus or any prospectus supplement or post-effective amendment
       to the registration statement has been filed and, with respect to the
       registration statement or any post-effective amendment, when the same has
       become effective;

              (ii)  of any written request by the Commission for post-effective
       amendments or supplements to the registration statement or prospectus;

              (iii) of the notification to Alliance by the Commission of its
       initiation of any proceeding with respect to the issuance by the
       Commission of, or the issuance by the Commission of, any stop order
       suspending the effectiveness of the registration statement; and

              (iv)  of the receipt by Alliance of any notification with respect
       to the suspension of the qualification of any Registrable Securities for
       sale under the applicable securities or blue sky laws of any
       jurisdiction.

       5.02   As soon as possible following receipt of notice from the Managing
Underwriter that a particular registration is a Cutback Registration, Alliance
will notify the Investor and each of the holders of Registrable Securities that
is a Significant Holder requested to be included therein that such registration
is a Cutback Registration and of the effect thereof on the ability of such
holders to include their shares in such registration.

       5.03   Alliance will use its best efforts to become (and thereafter to
remain) eligible to effect Short-Form Registrations.

       5.04   Whenever the holders of Registrable Securities have requested that
any Registrable Securities be registered pursuant to Section 2 or Section 3,
each holder of Registrable Securities will be deemed to have agreed that, upon
receipt of 

                                      -13-

 
any notice from Alliance of the happening of any event of the kind described in
Section 5.01(e) or Section 5.01(k)(ii) or (iii), the holders of Registrable
Securities covered by such registration statement will forthwith discontinue
disposition of any such Registrable Securities until the holders of Registrable
Securities receive copies of the supplemented or amended prospectus contemplated
by Section 5.01(e), or until they are advised in writing by Alliance that the
use of the applicable prospectus may be resumed, and they have received copies
of any additional or supplemental filings that are incorporated or deemed to be
incorporated by reference in such prospectus (it being the agreement of the
parties hereto, however, that the obligation of Alliance with respect to
maintaining the subject registration statement current and effective shall be
extended by a period of days equal to the period the holders of Registrable
Securities are required by this Section 5.04 to discontinue disposition of such
Registrable Securities.

SECTION 6.    Indemnification; Contribution.

       6.01   With respect to any registration, qualification or compliance
effected or to be effected pursuant to this Agreement, Alliance shall indemnify
each holder of Registrable Securities whose securities are included or are to be
included therein, each such holder's directors, officers, employees,
stockholders, Affiliates and agents, each underwriter (as defined in the
Securities Act) of the securities sold by such holder and each Person who
controls (within the meaning of the Securities Act) any such holder or
underwriter, from and against all claims, losses, damages and liabilities (or
actions in respect thereof) arising out of or based on:

              (i)   any untrue statement (or alleged untrue statement) of a
       material fact contained in any prospectus, offering circular or other
       document (including without limitation any related registration
       statement, notification or the like), or any amendment thereof or
       supplement thereto, incident to any such registration, qualification or
       compliance;

              (ii)  any omission (or alleged omission) to state therein a
       material fact required to be stated therein or necessary to make the
       statements therein not misleading; or

              (iii) any violation by Alliance of the Securities Act, the
       Exchange Act or any rule or regulation promulgated thereunder applicable
       to Alliance, or of any blue sky or other state securities laws or any
       rule or regulation promulgated thereunder applicable to Alliance,

                                      -14-

 
and will reimburse each such Person entitled to indemnity under this Section
6.01 for all legal and other expenses reasonably incurred, as the same are
incurred, in connection with investigating or defending any such claim, loss,
damage, liability or action; provided, however, that the foregoing indemnity and
reimbursement obligation shall not be applicable to the extent that any such
claim, loss, damage or liability arises out of or is based on any untrue
statement (or alleged untrue statement) or omission (or alleged omission) or
violation made in reliance upon and in conformity with written information
furnished to Alliance by such holder specifically for use in such prospectus,
offering circular, other document, amendment or supplement; and provided further
that the foregoing indemnity and reimbursement obligation shall not be
applicable with respect to any preliminary prospectus to the extent that any
loss, claim, damage, liability or expense of the indemnitee results from the
fact that a holder of Registrable Securities sold Registrable Securities to a
person to whom there was not sent or given, at or prior to the written
confirmation of the sale, a copy of the prospectus (excluding documents
incorporated by reference) or of the prospectus as then amended or supplemented
(excluding documents incorporated by reference) if Alliance has previously
furnished copies thereof to the holder of Registrable Securities in compliance
with Section 5 of this Agreement and the loss, claim, damage, liability or
expense of the indemnitee results from an untrue statement or omission of a
material fact contained in such preliminary prospectus which was corrected in
the prospectus (or the prospectus as amended or supplemented).

       6.02   With respect to any registration, qualification or compliance
effected or to be effected pursuant to this Agreement, each holder of
Registrable Securities which are included or are to be included in such
registration, qualification or compliance shall indemnify Alliance, its
directors, officers, employees, stockholders, Affiliates and agents, each
underwriter (as defined in the Securities Act) of the securities of such holder,
each Person who controls (within the meaning of the Securities Act) Alliance or
any such underwriter from and against all claims, losses, damages and
liabilities (or actions in respect thereof) arising out of or based on:

              (i)  any untrue statement (or alleged untrue statement) of a
       material fact contained in any prospectus, offering circular or other
       document (including without limitation any related registration
       statement, notification or the like), or any amendment thereof or
       supplement thereto, incident to any such registration, qualification or
       compliance;

              (ii) any omission (or alleged omission) to state therein a
       material fact required to be stated therein or necessary to make the
       statements therein not misleading;

                                      -15-

 
              (iii) any violation by such holder of the Securities Act, the
       Exchange Act or any rule or regulation promulgated thereunder applicable
       to such holder, or of any blue sky or other state securities law or any
       rule or regulation promulgated thereunder applicable to such holder, and
       will reimburse each such Person entitled to indemnity under this Section
       6.02 for any legal and other expenses reasonably incurred in connection
       with investigating or defending any such claim, loss, damage, expense,
       liability or action; or

              (iv)  with respect to any preliminary prospectus, the fact that 
       any holder of Registrable Securities sold Registrable Securities to a
       person to whom there was not sent or given, at or prior to the written
       confirmation of the sale, a copy of the prospectus (excluding documents
       incorporated by reference) or of the prospectus as then amended or
       supplemented (excluding documents incorporated by reference) if (a)
       Alliance has previously furnished copies thereof to the holder of
       Registrable Securities in compliance with Section 5 of this Agreement and
       (b) the loss, claim, damage, liability or expense of the indemnitee
       results from an untrue statement or omission of a material fact contained
       in the preliminary prospectus which was corrected in the prospectus (or
       the prospectus as amended or supplemented);

but in each case of the preceding subsections (i), (ii), (iii) and (iv), only to
the extent that such untrue statement (or alleged untrue statement) or omission
(or alleged omission) or violation is made in such prospectus, offering
circular, other document, amendment or supplement in reliance upon and in
conformity with written information furnished to Alliance by such holder
specifically for use in such prospectus, offering circular, other document,
amendment or supplement.

       6.03   Each Person entitled to indemnification under this Section 6 (an
"Indemnified Party") shall give notice to the party required to provide
indemnification (the "Indemnifying Party") promptly after the Indemnified Party
has actual knowledge of any claim as to which indemnity may be sought, and shall
permit the Indemnifying Party to assume the defense of any such claim or any
litigation resulting therefrom; provided, however, that:

              (i) counsel for the Indemnifying Party who shall conduct the
       defense of any such claim or any litigation shall be approved by the
       Indemnified Party (which approval shall not be unreasonably withheld or
       delayed);

                                      -16-

 
              (ii)  the Indemnified Party may participate in such defense at the
       Indemnified Party's expense; provided, however, that the Indemnified
       Party or Indemnified Parties shall have the right to employ a single law
       firm and a single local counsel law firm to represent it or them if, in
       the reasonable judgment of the Indemnified Party or Indemnified Parties,
       it is advisable for it or them to be represented by separate counsel by
       reason of having legal defenses which are different from or in addition
       to those available to the Indemnifying Party, and in that event the
       reasonable fees and expenses of one such law firm and one such local law
       firm shall be paid by the Indemnifying Party; and

              (iii) failure of any Indemnified Party to give notice as provided
       herein shall not relieve the Indemnifying Party of its obligations under
       this Section 6.

No Indemnifying Party, in the defense of any such claim or litigation, shall,
except with the consent of the Indemnified Party to which such claim or
litigation relates, consent to entry of any judgment or enter into any
settlement unless such settlement relieves the Indemnified Party of any and all
liability.  Each Indemnified Party shall furnish such information regarding
itself for the claim in question as an Indemnifying Party may reasonably request
in writing and as shall be reasonably required in connection with defense of
such claim in litigation resulting therefrom.

       6.04   If the indemnity and reimbursement obligation provided for in each
of Section 6.01 and Section 6.02 is unavailable or insufficient to hold harmless
an Indemnified Party in respect of any claims, losses, damages or liabilities
(or actions in respect thereof) referred to therein, then the Indemnifying Party
shall contribute to the amount paid or payable by the Indemnified Party as a
result of such claims, losses, damages or liabilities (or actions in respect
thereof) in such proportion as is appropriate to reflect the relative fault of
the Indemnifying Party, on the one hand, and the Indemnified Party, on the other
hand, in connection with statements or omissions which resulted in such claims,
losses, damages or liabilities, as well as any other relevant equitable
considerations.  The relative fault shall be determined by reference to, among
other things, whether the untrue or alleged untrue statement of a material fact
or the omission or alleged omission to state a material fact relates to
information supplied by the Indemnifying Party or the Indemnified Party and the
parties' relative intent, knowledge, access to information and opportunity to
correct or prevent such untrue statement or omission.  The parties hereto agree
that it would not be just and equitable if contributions pursuant to this
Section 6.04 were to be determined by pro rata allocation or by any other method
of allocation which does not take account of the equitable considerations
referred to in the first sentence of this 

                                      -17-

 
Section 6.04. The amount paid by an Indemnified Party as a result of the losses,
claims, damages or liabilities referred to in the first sentence of this Section
6.04 shall be deemed to include any legal and other expenses reasonably incurred
by such Indemnified Party in connection with investigating or defending any
claim, loss, damage, liability or action which is the subject of this Section
6.04.

       No Indemnified Party guilty of fraudulent misrepresentation (within the
meaning of Section 11(f) of the Securities Act) shall be entitled to
contribution from the Indemnifying Party if the Indemnifying Party was not
guilty of such fraudulent misrepresentation.

       The provisions of this Section 6 shall be in addition to any other rights
to indemnification or contribution which an indemnified party may have pursuant
to law, equity, contract or otherwise and shall remain in full force and effect
regardless of any investigation made by or on behalf of an Indemnified Party and
shall survive the transfer of the shares of Ordinary Shares or other stock or
securities which may be issued upon exercise of the Warrants.

SECTION 7.  Information by Holders.

       If Registrable Securities owned by a holder are included in any
registration, such holder shall furnish to Alliance such information regarding
itself and the distribution proposed by such holder as Alliance may reasonably
request and as shall otherwise be required in connection with any registration,
qualification or compliance referred to in this Agreement.

SECTION 8.  Rule 144 Reporting; Rule 144A Sales.

       With a view to making available to each holder of Registrable Securities
the benefits of certain rules and regulations of the Commission which may permit
the sale of the Registrable Securities to the public without registration,
Alliance agrees that until the earlier of (a) the date on which no holder owns
any Registrable Securities or (b) the Expiration Date:

              (a) Alliance shall, at any time after any of Alliance's securities
       are registered under the Securities Act or the Exchange Act: (i) make and
       keep available public information, as those terms are contemplated by
       Rule 144; (ii) timely file with the Commission all reports and other
       documents required to be filed under the Securities Act and the Exchange
       Act; (iii) furnish to each holder of Registrable Securities forthwith
       upon request a written statement by Alliance as to its compliance with
       the reporting 

                                      -18-

 
       requirements of the Securities Act and the Exchange Act, and a copy of
       the most recent annual or quarterly report of Alliance; and (iv) comply
       with all rules and regulations of the Commission applicable in connection
       with the use of Rule 144 and take such other actions and furnish such
       holder with such other public information as such holder may reasonably
       request in order to assist such holder in availing itself of any rule or
       regulation of the Commission allowing such holder to sell any Registrable
       Securities without registration; and

              (b) each holder of Registrable Securities and each prospective
       holder of Registrable Securities who may consider acquiring Registrable
       Securities in reliance upon Rule 144A shall have the right to request
       from Alliance, and Alliance will provide upon request, such public
       information regarding Alliance and its business, assets and properties,
       if any, as such holder may reasonably request so as to assist such holder
       in the transfer of Registrable Securities to such prospective holder in
       reliance upon Rule 144A.

SECTION 9.  Other Registration Rights.

       9.01   Alliance represents and warrants to the Investor that there is not
in effect on the date hereof any agreement by Alliance (other than this
Agreement and the other Registration Agreements) pursuant to which any holders
of securities of Alliance have a right to cause Alliance to register or qualify
such securities under the Securities Act or any applicable state securities
laws.

       9.02   So long as any Registrable Securities shall be outstanding, (a)
Alliance shall not amend or permit the amendment of the Registration Agreements
in any manner that is inconsistent with this Registration Rights Agreement or
which adversely affect the rights of any holder of Registrable Securities
without the prior written consent of the holders of a majority of the then
outstanding Registrable Securities and (b) Alliance shall send any notice in
respect of a registration to be delivered by Alliance to any holder of any
rights under any of the Registration Agreements to the Investor and any
Significant Holders of Registrable Securities.  So long as any Registrable
Securities shall be outstanding, prior to the Expiration Date (as defined in the
Warrant) Alliance shall not agree with the holders of any securities issued or
to be issued by Alliance to register or qualify such securities under the
Securities Act or any applicable state securities laws unless such agreement
(including any Registration Agreement) specifically provides that: (a) such
holder of such securities may not participate in any Piggyback Registration
except as provided in Section 2; and (b) the holder of such securities may not
participate in any Requested Registration except as provided in Section 3.

                                      -19-

 
SECTION 10.  Holdback Agreements.  In order to facilitate the possibility of
future public offerings of Ordinary Shares, the holders of Registrable
Securities agree that the Registrable Securities will not be resold during a
period commencing on the filing by Alliance of a registration statement under
the Securities Act for an underwritten public offering for cash by Alliance of
Ordinary Shares or securities convertible into or exercisable or exchangeable
for its Ordinary Shares and continuing until the earlier of the abandonment of
the proposed public offering or 120 days following the date of the last closing
in the public offering without the consent of the underwriters of such offering,
except to the extent such shares are included in such registration.  Holders of
such Registrable Securities also agree that they will cooperate with Alliance in
providing reasonable written assurances respecting the foregoing to the
underwriter of any such public offering.  Holders agree that during the above
restricted period they will not directly or indirectly sell, offer to sell,
contract to sell (including without limitation any short sale), grant an option
to purchase or otherwise transfer or dispose of (other than to donees who agree
to be similarly bound) shares of Registrable Securities at any time during such
period except securities included in such registration.  In order to enforce the
foregoing covenant, Alliance may impose stop-order instructions with respect to
such shares of Registrable Securities held by each holder, which shall be
binding upon any assignee or successor of such holder (and the shares or
securities of every other person subject to the foregoing restriction), until
the end of the restricted period.

SECTION 11.  Miscellaneous.

       11.01  Successors and Assigns.  Subject to the provisions of Section 13,
this Agreement shall inure to the benefit of and shall be binding upon the
parties hereto, all the holders of Registrable Securities and their respective
legal representatives, successors and assigns.

       11.02  Severability.  If any term or provision of this Agreement, or the
application thereof to any Person or circumstance, shall, to any extent, be
invalid or unenforceable, the remaining terms and provisions of this Agreement
or application to Persons and circumstances shall not be invalidated thereby,
and each term and provision hereof shall be construed with all other remaining
terms and provisions hereof to effect the intent of the parties hereto to the
fullest extent permitted by law.

       11.03  Notices.  All notices, requests and other communications provided
for herein (including without limitation any waivers or consents under this
Agreement) shall be sent in accordance with Section 7.02 of the Warrant
Agreement.

                                      -20-

 
       11.04  Certain Terms.  As used herein, the neuter gender shall also be
deemed to denote both the masculine and feminine genders.  Unless the context
otherwise requires, the words "hereof", "herein", "hereto" and "hereunder", and
words of similar import, when used in this Agreement shall refer to this
Agreement as a whole and not to any particular term or provision of this
Agreement.  Whenever the context requires, the singular form of any noun,
pronoun or verb includes the comparable plural form thereof, and vice versa.

       11.05  Counterparts.  This Agreement may be executed with counterpart
signature pages or in several counterparts which, when executed and delivered by
all parties hereto, shall be binding on all parties hereto and shall constitute
one Agreement, notwithstanding that all parties have not signed the same
signature page or the same counterpart.

       11.06  GOVERNING LAW.  THIS AGREEMENT SHALL BE CONSTRUED AND ENFORCED IN
ACCORDANCE WITH, AND SHALL BE GOVERNED BY, THE LAWS OF THE STATE OF ILLINOIS
APPLICABLE TO CONTRACTS EXECUTED IN AND TO BE FULLY PERFORMED IN SUCH STATE.

       11.07  Captions.  The headings in this Agreement are for purposes of
reference only and will not be considered in construing this Agreement.

       11.08  Amendments, Waivers, etc.  This Agreement may be amended only by a
written instrument (which may be executed in any number of counterparts) signed
by Alliance and the holders of a majority of the Registrable Securities voting
as a class; provided, however, that no such amendment, without the consent of
all holders of Registrable Securities at the time outstanding, shall amend this
                                                                               
Section 11.08.  Subject to Section 11.09, no provision of this Agreement may be
waived except by a written instrument signed by the party hereto sought to be
bound.  No failure or delay by any party hereto in exercising any right or
remedy hereunder or under applicable law will operate as a waiver thereof, and a
waiver of a particular right or remedy on one occasion will not be deemed a
waiver of any other right or remedy, or a waiver on any subsequent occasion.

       11.09  Consents of Holders of Registrable Securities.  Any consent of the
holders of Registrable Securities pursuant to this Agreement, and any waiver by
such holders of any provision of this Agreement, shall be in writing (which may
be executed in any number of counterparts) and may be given or taken by the
holders of a majority of the Registrable Securities voting as a class; provided,
however, that no such consent or waiver, without the consent of all holders of
Registrable Securities 

                                      -21-

 
at the time outstanding, shall amend this Section 11.09; and any such consent or
waiver so given or taken will be binding on all the holders of Registrable
Securities.

       11.10  Recapitalization, Exchanges, etc., Affecting Alliance's Capital
Stock. The provisions of this Agreement shall apply, to the full extent set
forth herein with respect to any and all shares of capital stock of Alliance or
any successor or assign of Alliance (whether by merger, consolidation, sale of
assets or otherwise) which may be issued in respect of, in exchange for or in
substitution of, the Registrable Securities and shall be appropriately adjusted
for any stock dividends, splits, reverse splits, combinations, recapitalizations
and the like occurring after the date hereof.

       11.11  Delay of Registration.  No holder of Registrable Securities shall
have any right to obtain or seek an injunction restraining or otherwise delaying
any such registration as the result of any controversy that might arise with
respect to the interpretation or implementation of this Agreement.

SECTION 12.  Listing on Securities Exchanges, etc.

       Alliance shall, promptly after the registration and sale thereof, use its
best efforts to cause any Registrable Securities (a) to be listed on a national
securities exchange and on each additional national securities exchange on which
similar securities of Alliance are listed, if the listing is then permitted
under the rules of such exchange, or (b) to be designated as National
Association of Securities Dealers Automated Quotation System ("NASDAQ")
"national market system securities" within the meaning of Rule llAa2-1 under the
Exchange Act if similar securities of Alliance are so designated.

SECTION 13.  Limitation on Registration Rights.

       Notwithstanding anything to the contrary contained herein, (a) the rights
of a holder of Registrable Securities under Section 3 hereof shall be terminated
on the tenth anniversary of the date hereof, (b) the rights of a holder of
Registrable Securities under Section 2 hereof shall be terminated on the tenth
anniversary of the date hereof, (c) prior to such time registration rights under
Sections 2 and 3 may be transferred only to transferees that, together with
their respective Affiliates, are Significant Holders (after giving effect to all
such transfers) and (d) no transfer of registration rights under Sections 2 and
3 may be made except in accordance with the terms and conditions set forth
herein, in the Warrant, and in the Warrant Agreement.

                                      -22-

 
       In addition, the rights and obligations under this Agreement shall
automatically be transferred to and binding on any transferee or assignee of the
Registrable Securities, provided that, such transferee or assignee:

              (i)   notifies Alliance in writing, within a reasonable time after
       such transfer, of the name and address of such transferee or assignee and
       the Registrable Securities with respect to which such registration rights
       are being transferred or assigned,

              (ii)  agrees in writing to be bound by and subject to the terms 
       and conditions of this Agreement,

              (iii) receives the Registrable Securities in a transaction that is
       in compliance with the requirements for an exemption from the
       registration requirements of the Securities Act and applicable state
       securities laws;

              (iv)  immediately following such transfer, is subject to
       restrictions under the Securities Act on further disposition of such
       Registrable Securities; and

              (v)   acquires at least 33% of the Registrable Securities the
       Investor is entitled to acquire under the Warrant Agreement.

                                      -23-

 
       IN WITNESS WHEREOF, the parties hereto have duly executed this
Registration Rights Agreement as of the date first written above.

                                        ALLIANCE RESOURCES PLC


                                        By:
                                           ------------------------------------
                                           Name:  Francis M. Munchinski
                                           Title: Assistant Corporate Secretary


                                        LASALLE STREET NATURAL RESOURCES 
                                        CORPORATION


                                        By
                                           ------------------------------------
                                           Name:  John H. Homier
                                           Title: President

                                      -24-

 
                                   EXHIBIT A
                                      to
                         Registration Rights Agreement

1.     Registration Rights Agreement by and among Alliance Resources PLC, a
corporation formed under the laws of England and Wales, and F. Fox Benton, Jr.,
Lizinka M. Benton, F. Fox Benton III, Lizinka C. Benton and Lucia T. Benton,
dated as of the 30th day of October, 1998.

2.     Registration Rights Agreement by and among ALLIANCE RESOURCES PLC, a
public limited company organized under the laws of England and Wales, ENCAP
EQUITY 1996 LIMITED PARTNERSHIP, a Texas limited partnership, and ENERGY CAPITAL
INVESTMENT COMPANY PLC, an English investment company, dated as of October 30,
1998.

                                      -25-