What the halacha is, is subject to debate. Some people argue that the halachic ban on polygamy should not apply and so far that is not a widely held position. If the United States were to formally recognize polygamous marriages, the halachic debate would become much more “interesting”. At present, America does not consider a second wife (with the first wife still married to the husband) as anything more than the husband’s “girl friend”, with no legal status at all.
Under American law, the first marriage is valid and the second marriage is consider void (i.e. only the first wife can be covered on a joint tax return, and she inherits the spousal share, and only the first wife can own a house as a tenant by the entireties.). However since American law no longer criminalizes consensual intimate relationships, it isn’t clear that criminal bigamy laws would be upheld, complicated by the issue as to whether this is a religious freedom issue.