In theory, all Jews are subject to the same Torah laws. In reality, there are different interpretations, traditions and history have informed those laws, and contrary to the assumptions of many Americans, there is a great deal more variety in Jewish tradition then Reform/Conservative/Orthodox.
The Torah (first five books of what Christians call the Jewish Scripture or Old Testament) is very clear on allowing polygyny (multiple wives) under specific conditions. Among other things, a man must be able to provide all his wives with the same level of care when it comes to their physical, and sexual needs, and he must be equally affectionate/loving/caring with all of them. Additionally, his first wife must agree before he can take a second wife, the first two must agree before he can take a third, and etc.
Additionally, the specifics of Torah law define adultery as a man sleeping with a woman who is married to someone else. He can sleep with as many unmarried women as he wants, and it’s frowned on, but not adulterous. Presumably the reverse is true (as long as a woman is unmarried she can sleep with any man she wants). Once the woman is married, it is adultery and against the law for both parties.
By extension polyandry (one woman with multiple husbands) would be considered adultery and is against Torah law.
Now, getting to those varieties of Judaism – some Jewish traditions consider the decisions of what is known as the Rabbinic period to be law one step below Torah law in importance. (Think of violating Torah law as on the level of murder and Rabbinic law as on the level of accidental manslaughter if that helps). Those who follow Rabbic law include the Ashkenazim (basically all the Jewish traditions Americans are familiar with) and some (but not all) of the Sephardi (Jews who lived in Spain, Portugal and the Ottoman Empire). Mizrahi (Jews from the Mi+ddle East or Asia) generally do not accept Rabbinic law.
This is important because around 1100 CE, it was decided by the Rabbis that polygyny would no longer be acceptable (several sources say that this was because the cultures they were living among didn’t practice polygyny, and it was dangerous to stand out too much). Regardless of the reason, at this point polygyny is considered illegal by the majority of Jewish traditions in Europe and America based on these Rabbinic laws.
Modern Jewish authority is pretty much against polygamy, and Israeli law says that Jews who married polygynously in their old countries can move to Israel, but once they are there they aren’t allowed any more polygynous marriages.This isn’t much of an issue as for practical reasons of civil laws even Jews whose traditions allow polygyny rarely practice it today.
Over all, from a technical reading of the Scripture, polyamory (multiple loves) is fine, because as long as the women aren’t married, it isn’t adultery. Similarly, a mono/poly marriage where the woman is mono and the man is poly could slip through the cracks as the man can have relations with women who are not married and it isn’t adultery.
However, multiple marriages of any variety are pretty much out.
All this depends on a very legalistic interpretation of the Scripture, and most Jewish communities frown on sex outside of marriage, even if it is technically legal. However, an important tenet of Judaism is that each person is responsible for their own interpretation of the law and how they follow the law.