My husband of 12 years has asked me for a divorce so that he can marry his mistress, a woman from Honduras, whom he has had a child with. I don't believe continuing the marriage is possible. But I remember seeing a story about a woman who sued her ex-husband's mistress for loss of companionship or something to that effect. Is this really possible?

Its possible to sue, its highly unlikely to win.

Could the lawsuit prohibit her entering the country? I probably know the answer already, but I thought I'd ask.

Not likely

It will have little effect on her entering the country. Your fight is at divorce.

Also the trick to the law suit, is how rich is she. If she has millions of dollars, you may collect, if she is poor, even if you win, you get nothing.

Most of the laws for adultry, and mistress , even if on the book are not enforceable due to Supreme Court cases over the last 15 years, that has taken Government our of the bedrooms.

... Supreme Court cases
over the last 15 years, that has taken Government our of the bedrooms.

Well, not exactly. Government is figuratively "in our bedrooms" now more than ever. It tries to dictate what kind of light bulb we use in our bedrooms, what fuel we heat our bedroooms with, how our bedrooms are constructed, and even requires labels on the matresseses we have in our bedrooms. What your metaphor suggests, of course, is the notion that government no longer regulates what sort of sex we have in or bedrooms, and with whom. This is true only to some extent. Nonconsensual sexual activity, sex with minors, etc. is still regulated.

But a suit for "alienation of affections" will no longer get anywhere in most areas of the U.S. I don't know about Honduras. If he seeks, however, a divorce from OP in the U.S. she can simply refuse. He will have to file the suit, but then he can get a judge to award him the divorce whether she likes it or not. Many years ago, if a man who had committed adultery wanted a divorce he would need his wife to file; he could not allege his own adultery as grounds for the divorce. Now, however, one doesn't need "grounds" for a divorce; "no fault" divorce makes "grounds" (other than irreconciliable differences - which either spouse can get the court to find without a need to actually prove it) unnecessary.