They always call it polygamy, but technically, if it's one man with multiple wives, it should be called polygyny. I know, I'm splitting hairs ...
So from what you're saying, the obstacles appear to be threefold:
- religious beliefs,
- lack of consent,
- social pressure.
Sounds a little like a lot of the men there are chafing for a religiously-sanctioned system of patriarchal polygyny. It existed in Old Testament times; the Mormons had it; why can't we have it? so to speak.
By the way, most (not all) polyamorists are of of the opinion that "polyamory without consent" isn't really polyamory at all; it's just cheating. So even if these wives know about their husbands' girlfriends, if said wives are (in their hearts) opposed to it, then the men are really just conducting a collection of affairs. Consent is usually considered a really important part of genuine polyamory.
I also want to add my personal feeling that it sounds pretty unfair that the men can have extra women, but the women can't have extra men. Of course, that was always unfair. It was unfair when the Old Testament prophets did it, and it's unfair when the Mormons do it. (I say "Mormon" loosely because the "main Mormon church" has given up the practice, however several "break-off Mormon sects" still do it.)
Religiously sanctioned patriarchal polygyny (commonly known as polygamy) is actually okay if the women really honestly deep down in their hearts consent to it. But the problem is, what if that religion resorts to lifelong brainwashing to get this "consent?" It's like saying it's okay to have sexual relations with a child if the child consents. The child doesn't have enough experience in life to give genuine informed consent.
I guess I kind of wandered off-topic there, but let's just say those are some of my thoughts about men having relations with multiple women.
I appreciate your input.