I don't think there's a particular term for a monarchy in this situation because there's no difference as to who is ruling. It's just a question of what title is granted to the spouse of the monarch.
In the United Kingdom and its predecessors (at least England), the title granted to the husband of a reigning Queen has never been King. King William III was the husband of the reigning Queen Mary II, but he wasn't King simply because of that, he was King because he was also co-ruler.
However, the title of the wife of a king is Queen. If Queen Elizabeth II were a man then she'd be King Elizabeth II, and the title of that king's wife would (generally) be Queen.
I say "generally" because Prince Charles's wife Camilla is expected not to take the title Queen should he be king. She is currently styled Duchess of Cornwall (he is the Duke of Cornwall) but not Princess of Wales (which she would be entitled to, but she and others felt the association of the title with Diana to be too strong).
So I think there's a false assumption in your question, which is that there are monarchies that "need to have both a king and a queen". If that happens anywhere it isn't the typical case. So it's not a question of needing a queen vs. not needing one, it's a case of the monarch happening to be both male and married, or not.
Who knows, maybe the next difficult question for the royal protocol officers will be how to style the husband of a reigning king, or the wife of a reigning queen ;-)