They will greet you warmly, shaking your hand, maybe with their two hands.
In Lebanon a dispensation for such marriages can be obtained and they are not uncommon.
Although permitted under Muslim law, polygamy is generally regarded as both impractical and undesirable because of the additional economic burden it places upon the household and because of the personal complications it entails.
The act of an individual can impact the perception of the entire family by others.
The interests of the family are expected to supersede those of the individual and loyalty (such as preferential treatment) is shown to fellow family members.
It's just what happens when you live in a place that's heavy on discourse, and it's a good thing..you embrace it.