Buddy ... you'd have to be extremely stupid to attempt legal marriage more than once concurrently in any US State. That's the broad definition. Sexual cohabitation with any number of people could fall under common-law marriage, but in practice it'd be far more difficult to prosecute. It's why poly couples have to make all kinds of alternative legal arrangements to cover power of attorney among the set.
And again, might I underscore this is an alternate Earth with harems as the default and no one's breathed the word "marriage," let alone "monogamy." Who are you trying to impress?
I'm not to impress a who. I'm trying to impress a what upon all who read this thread:
Laws exist that can punish polygamy even without proof of plural marriage. Based solely on Polygamous cohabitation.
I don't know how strict rules are against political discussions are here, so I'm just trying to point out that such laws exist.
(Granted, in at least South Africa, there exist laws that insist on only acknowledging one marriage. But they probably don't have laws against adultery, at least not any that directly punish the culprits. So no indirect punishment is encountered by having only one legal wife.
I mention this because if anyone else wanted harems to exist, they would probably have to grapple with the constant threat of such laws being potentially enforced.