Worldwide, the acceptance of polygamy is the norm, not the exception. Murdock’s “Ethnograpic Atlas Codebook” surveyed pretty much all the cultures we had decent information on and correlated factors across cultures to see where differences and commonalities were. There’s a PDF that I found online here that shows the breakdown. Of the 1231 cultures for which data existed, only 186 were preferentially monogamous. Most practiced some form of polygamy, and 4 practiced some kind of polyandry.
That means that there’s no particular reason it can’t work. In fact, it does work in the vast majority of human societies. And like a couple of people in this thread recently said, marriage arrangements often come down to money. In the case of polygamy one reason that people get together in group marriages is because it makes economic sense. There are benefits for sharing work burdens, having a larger pool of support, safeguarding against disaster, among other reasons.
This paper discusses the economics of polygyny. It’s fairly technical (frankly, some of the math is over my head), though broad and not considered by the author to be an extensive treatment on the subject. The view that the only reason men marry multiple wives is to get hot second-wife sex is too simplistic. There are a lot of reasons social, economic, and practical, that polygamous marriage takes place.
The legal problems that people have pointed out are largely due to some assumptions about the benefits that are folded into marriage contracts. If those things were spelled out instead of being assumed, I think that it wouldn’t be particularly difficult to write marriage contracts that work for multiple marriages. After all, we’ve got examples from over 1000 cultures, including past examples even in European cultures, as to what works and what doesn’t.
Multiple divorces can often bring more legal entanglements than I think a good multiple-marriage contract would. It could be argued that what we’ve got now, serial monogamy, is an inferior form of polygamy. While a man in our society can’t be married to more than one woman at a time, he often has to support more than one household. His wives don’t get the benefit of of labor-sharing and de-facto daycare/babysitting from the co-spouse, and all of them don’t get the economic benefit of having a shared household with lower costs than multiple households.
The laws and customs of inheritance, alimony, child support, custody, and property division can get quite complex, and even worse have to be ironed out after the fact because there’s no provision in the marriage contract itself to handle any of that. The laws we have regarding the disposition of those things after marriage are tacked onto a system where the default used to be a lifelong contract, but many, perhaps most, marriages do not last for a lifetime.
It’s not like modern industrial society hasn’t dealt with group living either. An extreme example, kibbutzim could be considered a form of large-scale group marriage, as marriage itself is sometimes discouraged as being undesirable for the community.
If we’re ever going to have same-sex marriages, these kinds of difficulties will have to be wrestled with to some extent anyway. Might as well do it right and consider everything at once rather than have a bunch of ad hoc kludges hacked onto the system to handle cases as they come up.
Something to note is that even in cultures where it’s considered highly desirable to have more than one spouse, it’s not particularly common, for any number of reasons both personal and economic. So even if we came up with a completely workable solution and approved it tomorrow, there wouldn’t be a huge rush of people getting into group marriages. It would probably be like the majority of polygamous societies around the world; mostly monogamous, a few with two spouses, and a very few (usually around 2% or less) with more than two spouses. I seriously doubt there would be any great societal upheaval caused by allowing people to get married to more than one person, especially when by all indications a small minority would be the only ones doing it.