What if my religion conflicts with ethical non-monogamy?
It depends on the religion, the specific community, and your relationship to both. Many practitioners in religious traditions have found ways to hold both; others have had to choose. There is no universal answer, but there are people in most traditions who have worked it through.
Religious tradition and ethical non-monogamy intersect in a range of ways depending on the tradition, the specific community, and the practitioner's own relationship to both. Some traditions are explicitly compatible with non-monogamous family structures (some pagan traditions, some progressive Jewish and Christian communities, some Unitarian Universalist congregations). Others are explicitly incompatible at the doctrinal level. Most fall somewhere in the middle, with the practical compatibility depending heavily on the specific local community.
What polyamorous practitioners in religious traditions have written about. The tradition's teachings are usually one of several inputs to a practitioner's discernment — alongside lived experience, the wellbeing of the people involved, conscience, community wisdom, and reading of scripture or text. Some traditions support that kind of multi-source discernment; others insist on doctrinal authority overriding the others. The practitioner has to decide how much weight to give each input.
What working it through often looks like. Long conversations with clergy or community members about the specific situation. Reading practitioners in the tradition who have addressed non-monogamy from within (every major religion has at least some such writers). Finding the subset of the community that is open to the conversation versus the subset that isn't. Sometimes finding a more compatible community within the same tradition.
When the tradition and the practice cannot coexist. Some practitioners find that, after real engagement, their tradition's claims and their practice's reality cannot be held together. The honest version of that is also a workable answer — leave the tradition, find a different community, accept the loss. Others find they can hold both with the caveat that they are no longer in good standing in their tradition's eyes; that is also a workable answer if the practitioner can accept the standing question. Others find they have to leave the practice; that, too, is sometimes what people genuinely conclude.
What does not work. Trying to hold both without doing the work of integration, which produces a kind of low-grade ongoing dishonesty that erodes both the faith life and the relational life. Hoping the tradition will change in time; it might, but probably not on your timeline. Pretending the conflict is smaller than it is. The integration takes work and is worth doing seriously.