IMO, your red flag warning was accurate. I would not touch this situation with a 10 foot pole.
I was with a couple that on the outside looked pretty tight. It was understood that I would be joining them. I am quite fond of closed triads, so joining a couple was not an issue for me. At first everything was great. You know when it ceased being great? When the husband and I fell in love. It had been talked about - that was the point, everyone loving everyone. Right?
Apparently not. The wife changed the rules. All I gleaned was that the husband was supposed to love only her, and I was supposed to love only her. The husband and I were not supposed to love each other. (They are now divorcing.)
My point is, if you grow to love this man, and some other woman has control over if and when you can have sex, (not to mention whether or not you can love each other) it will cause you serious heartache.
As for why the rule, I don't have a clue. I know were my SO out on a date with someone, I would feel ridiculous if he called to ask permission.