Her husband allows the affair

As their marriage wasn't based on love, but instead arranged to further the continuation of the family line, Minami's husband wasn't exactly invested to taking care of her and ensuring her faithful to him. Just as he was pursuing another woman, he allowed you to do the same with Minami. He acknowledged your relationship

It wasn't a blessing or disapproval.

However, it will be up to you on you how to go forward with your illicit relationship. Will you continue your affair with Minami even if she remains legally wedded to another man and any child that results from it will not inherit your family name?