What would happen if a foreign polygamous couple resided in let's say New York? An opinion letter by the Maryland attorney general concerning recognition of out of state same sex marriages had this interesting tidbit:
Fn 53 While it remains extremely unlikely that Maryland – or any state – would recognize foreign polygamous marriages generally, such marriages have been recognized by American courts for specific purposes, such as inheritance and property succession. For example, in a case involving a native of India who died intestate in the United States, the court held that his two legally wedded wives in India would be allowed to share in the estate, and indicated that the public policy exception would have precluded recognition “only if decedent had attempted to cohabit with his two wives in California.” In re Dalip Singh Bir’s Estate, 188 P.2d 499, 502 (Cal. App. 1948); see also Scoles & Hay, Conflict of Laws (1984) at 446 (“The courts do recognize the legal existence of and give effect to foreign matrimonial unions that do not conform to requirements for the marriage relationship among their own people.... It may be doubted whether a foreign visitor would be permitted to cohabit here with his four wives, although even this is uncertain. Children of the union would probably be recognized as legitimate.")
The 1948 ruling cited a few cases from the 19th Century involving possibly polygamous Native American marriages. At least one case involved legal liability or rights not present if a woman was married; would this be transferable to a case of spousal immunity for multiple wives? California is more liberal in respect to comity, but cases from more conservative states were cited. As shown, it also is not just a matter that suddenly arose in our same sex marriage era.
In fact, given some tribes had same sex marriages for quite some time, that very issue in theory could have arose in the 19th Century as well. At least, applying general principles. Meanwhile, there is a case in Canada examining the very issue of polygamy. As noted by the letter, this is a potent issue given the number of nations that allow the practice.