Is it permissible in Islam to marry a man who is already married, with both parties aware of a previous romantic relationship, in a country that does not permit polygyny, such that the state would not recognize this marriage?

Islam permits polygyny, even in countries where it is not legally recognized, based on the Almighty's saying: (Then marry such women as seem good to you, two, three, or four). The questioner sister can marry this man in the presence of a guardian (wali), two witnesses, and the announcement of the marriage. It is not a condition for it to be officially registered or for the first wife to be aware of it. If this is not possible, the advice for her is to turn her heart away from this man, for there might be good in that, as the Almighty said: (And whoever fears Allah - He will make for him a way out) and (But if they separate [by divorce], Allah will enrich each of them from His abundance). She should increase her acts of worship, supplications, and humble entreaties to Allah.