They may claim they are their wives... but let's be honest here. They are catholic and as such are not permitted to truly have two wives.
The catholic church recognizes that as bigomy and that in no way connected to being legal or illegal in regards to immigration. They may be shitbags, but they certainly don't have "two wives".
of course, I was told by someone of another country, that "El Salvadorians don't marry the women they are with", this of course is probably cultural bias.
But I can tell you, dude will tell me, "I live with my wife", where is your wife, I say, she is in ______ (not the US). So this woman you are living with is not your wife, "No", do you have kids with this person, "Yes", so she is your girlfriend, not your wife, "Yes" .
This has happened more times than I can count. They say "My Wife", because they are living with this woman, like a wife. They of course don't put a ring on it. And as i said they will have a US "wife" and family here, but legally married in another coutry, with kids there. Catholic or not, there Catholicism by practice and Catholicism by "word".