MM, as ronsnr pointed out, there is a long history of anomosity in the US towards polygamy. This history makes it very unlikely that US laws against polygamy will be changed in our lifetimes.
As I stated in my previous post, authorities can indeed prosecute for bigamy if a man has a common-law marriage relationship with two different women, even if he is not married to both. Generally authorities won’t prosecute unless there are other issues such as underage marriage or welfare fraud, but laws exist allowing them to do so.