Don’t forget the Confessional Lutherans, the Assyrian Church of the East and the Ancient Church of the East. The latter two will permit NIcene Christians to receive the Eucharist, but only if they believe in the real presence, which I think is as open as any of the traditional liturgical churches that believes in the real presence can safely be.
The bitter irony in all of this is that the aforementioned churches with the possible exception of the Confessional Lutherans are among the least likely to bother you over any issue of natural science of evolution; indeed the idea of the BIg Bang as an explanation of Hubble’s discovery of the increasing distance between galaxies was articulated by a Roman Catholic priest who engaged in astronomy (and not even at the Vatican Observatory, if memory serves, although that is a very good scientific institute).
The doctrine you’re being asked to accept by the way as a precondition to receiving our Eucharist - not even to attending our services, mind you, or receiving a blessing, or in the case of the Orthodox, antidoron (blessed bread) but merely receiving the Eucharist, is that you believe what Christ our True God said about it, that it is His Body, broken for us and for many for the remission of sins, and His blood of the new covenant, and that the Eucharist is an anamnesis of our Lord, in which we participate in the Last Supper - there is indeed only one Eucharist, and one Baptism, and that is what we participate in.
Also in the case of the Orthodox an Orthodox Christian cannot be an Iconoclast, a Nestorian, a Eutychian, a Monothelite, or otherwise an adherent of a doctrine contrary to the faith of the ecumenical councils (the Oriental Orthodox did not participate in the fifth, sixth and seventh council and disagreed with the specific wording adopted at the fourth, but did agree with the fourth in terms of having anathematized Eutyches, and with the Sixth and Seventh also anathematized monothelitism and iconclasm despite not participating in the councils; indeed unlike the Eastern Orthodox the Oriental Orthodox never had a Patriarchate come under the control of Iconoclasts).
Even with regards to the Eucharist, what matters to the Eastern churches, with our emphasis on apophatic theology (using the via negativa rather than affirmative statements of doctrine) is that people do not believe a theology that denies the real physical presence of Christ; as far as how that presence occurs, we have no official explanation, other than it is a sacred mystery, although on occasion some of our saints have used the word Transubstantiation, but primarily to differentiate our position from that of Calvinists who admit only a spiritual presence, or Zwinglians who reduce the Eucharist to a sign or symbol, or Memorialists who deny even that, or Receptionists.
And there are scriptural reasons for not believing in Zwinglianism, Memorialism or Receptionism, for our Lord did not say “This is a symbol of my body” or “This is not really my body in any meaningful way” or “This will not become my body until you partake of it” but rather “This is my body” and “This is my blood.” Which is why Martin Luther famously carved HOC EST CORPVS MEVM into the table at the Marburg Colloquy.