Even in the UK and US, there are already quite a few stable polyamorous relationships, although they have no legal standing. I can't say I'm for or against it. It's a case of each to their own, but at least if it were legalised, it would give the partners rights to inherit etc. It would probably make divorce an absolute nightmare.
There are various different types of polyamorous relationships too. The most common format, as far as I know is one guy with multiple women, usually totally straight, but sometimes the women are involved with each other too. Some, it's a couple of guys who share a girl but not each other or sometimes it's a three-way. You'd have to legislate for that too. There's the option of one person with multiple partners who are not in relationships with each other, ie each has an individual marriage to the same person. Another option would be that they're all married to each other. What if one partner wants to add a third or fourth? What if two want to kick the rest out of the relationship and go exclusive? I can see it turning into an absolute minefield where the only people benefiting would be lawyers. Add kids to the mix and it gets even worse.
From what I understand of Muslim countries where polygamy is practised, it's all about the man. He's allowed to have up to four wives and they are fully dependent on him. He can divorce them, but if they have children, he has a responsibility to pay for their upkeep. I think, but I could be wrong, that what usually happens is that the divorced wife stays on in the household to raise her children unless she marries another man, in which case the kids stay with their father and his other wives. Please correct me if I'm mistaken on that.