Let's get this out of the way up front- I am a lawsperg, not a lawyer, and even if I were a lawyer estate law is a messy patchwork of federal and state laws. If you want a better explanation ask AnOminous but don't be too surprised if he just says poly is for cucked retards.
I'm going to start with death rather than a breakup because it's simpler. When someone dies without a will, if there is an estate after creditors' claims are satisfied it goes to the next of kin - children get first priority, then spouses, then parents, then siblings, then extended family. Many poly couples are not legally married because no state in the US recognizes polygamy/bigamy and spouses are only considered next of kin if they're legally married, so a poly bf/gf has no more legal right in an estate dispute than any other random person off the street. So unless there is a will, or assets are in joint accounts, any estate dispute with a poly couple will leave the survivors with nothing. Since most people into poly are extremely dysfunctional and childish I feel reasonably confident that estate planning, wills, or even just setting up a joint bank account to simplify things are not going to cross their minds.
For communal property in a breakup, since plural marriage is not recognized, there is no legal issue at all if the person leaving is either not on the deed/title/mortgage/car loan or the only person on there. This is true even if the person leaving helped pay for the property. On the other hand, if the property is in multiple people's names or there's nothing to establish ownership because there is no deed/title/whatever (so pretty much anything other than a car, home, or maybe luxury goods) then there is no cut-and-dry procedure even if there's no debate over who paid for or used the property the most. If the tards can't figure it out on their own they'd have to go to court and have a judge or mediator sort things out.
Edit: I forgot to mention common law marriage before and that would probably be inapplicable: Common law marriage governs two people who live together and act in public like a married couple. The first bit, "two people" disqualifies any poly relationship unless it's a husband and wife who date other people on the side (i.e. there is an obvious distinction between the couple and everyone else involved). Even then this is something courts probably have never considered before so I don't feel at all confident guessing whether they'd consider that common-law (but it would be a great candidate for our lolsuit board).
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