One issue with polyamorous marriage that I could see is the issue of divorce. If one of the partners divorced, what portion of alimony would go to the divorced? Say there are four people in a relationship, persons w, x, y all married to person z. Under normal circumstances, if there was a divorce, the alimony would usually be split 50/50, but that isn't possible in this situation. It would have to be proportionate to the number of people contracted to one person, so people w, x and y would only get 25%.
Then comes the issue if contracts are between different people, say people w and x are married and x is married also to person z, and person z is also married to person y, while y is also married to person w. Yes, calculations can work it out evenly, but it would be a paperwork nightmare. Like said by Vararam, there would need to be a huge overhaul with how marriage is handled.