Polygamy is legal in some countries and illegal in some others. What is the legal status of a polygamic family in a monogamic country? If this is not regulated by international law let's narrow this down to France.
Specifically:
who is considered the "wife" or "husband" (in the case of polyandry) in such a family? Is this someone designated automatically (say, the first marriage) or designated by the single "half" of the family?
what is the effective status of children in these families?
It's not regulated by international law.
Depending on the laws of the intended destination country, it may be the case that none of the members of the family are permitted to enter, or that they can enter, but only two can enter as a "couple", while the others are legally completely separate (or even excluded from the country).
According to one blog, at present in the United States,
a foreign national must actually intend to practice polygamy in the United States to be ineligible for an immigrant visa. The current law does not prevent a polygamist or someone who practiced polygamy in the past or expresses a belief in polygamy from being eligible for an immigrant visa. But aliens coming to the United States to practice polygamy are barred.
Before 1990, there was a law on the books by which someone who merely "advocated the practice of polygamy" could have been barred.
This question has been answered on Quora for the U.S., again for the U.S. (where the question asked about "US and EU"), and for Canada. Basically, the whole family can't legally immigrate as a unit. However, determining who is the "real" wife, if any of them, would depend on the facts of the case, the specific laws of the target jurisdiction, and the purpose of the determination. It could very well be that every one of the other marriages would void a new marriage in the destination country and entitle the children to child support, but none of them would entitle the wife to a spouse's visa or the father to visitation rights after a purported divorce.
See also this answer about whether it's possible for a married immigrant to commit bigamy by entering the United States pretending to be unmarried. Sure there are people who try, and it's more likely to succeed with the cooperation of the foreign spouse(s), but it's against the law and can be grounds for deportation, imprisonment, annulment of the second marriage, or exclusion of any polyspouse who is outside the country.
In Australia, the first marriage is recognised as a marriage, the subsequent marriages are not (s88D of the Marriage Act 1961).
The status of children in Australia is unaffected in any way by the marital status of their parents.
