A very kind reader sent along a description of the legal theory by which the polygamous freedom to marry can be accomplished, and I will offer my thoughts as well. It's a long essay, Ultimately, the thing to take away is that our laws can accommodate people in any form of polyamory who want legal recognition of a marriage (polyagamy). Since as far as the government involvement is concerned, marriage is mostly about financial and "who has responsibility for whom" issues, contract and business law has already demonstrated that three or more people can have legally recognized relationships. I had previous written about this here.
There is an ongoing discussion among polyamory activists regarding a legal model of polyamorous marriage (i.e., the extension of the legal concept of marriage to include polyamorous families). One debate centers around the relative merits of an all-with-all approach to marriage (whereby three or more persons are all joined together at the same time within a single marriage) and dyadic networks (whereby existing laws against bigamy are revised such that people are perfectly free to be concurrently married to multiple other persons, provided that each such new marriage is preceded by a legal notification regarding the pending new marriage to all those to whom one is already married; failure to provide that legal notification would then constitute the updated crime of bigamy).
I think both should be offered. The basic paperwork can actually be rather simple.
Dyadic networks would result in what might be thought of as a "molecular" family structure — one which might be best represented by the molecular diagrams commonly used in chemistry. In this way, marriage would remain a dyadic relationship (i.e., a relationship between two persons), thus minimizing any changes to the existing system of legal marriage, but the introduction of concurrency would provide access to legal marriage for polyamorous families.
Dyadic networks can correctly represent any situation associated with the "all-with-all" paradigm, as well as many situations that the "all-with-all" paradigm cannot deal with. A "complete" dyadic network would take the form of a complete graph, in which every person is (pairwise) married to every other person, thus correctly representing any situation associated with the "all-with-all" paradigm.What this is saying that if A, B, and C all want to marry as a triad, with a dyadic model that can be accomplished through A & B marrying, A & C marrying, and B & C marrying. However, the all-with-all model automatically does this with one ceremony and one piece of paper. All-with-all wouldn't be what every polycule would want, as detailed below. It would only be a desirable option for polycules in which every individual wants to be married to all of the others. It would only work for the triangle or the square below.