In OT times girls were property that were raised and sold for a profit. Their value was in their virginity. When a man bought a girl she then became his property. Like it or not, that's the way it was and that was OT marriage. In fact, if you think about it that way, adultery in the OT was sort of a type of theft! At any rate, my point is that men wouldn't really have a sexual relationship with what we think of as a girlfriend because no father (owner) would allow it. If you wanted to have sex with her, you paid for her and married her. So when we see OT polygamy it probably is the closest parallel to the hypothetical girlfriend question you asked.
Now as for my original statement here, gee, I didn't intend to have such a reaction. I feel like such a troll. Somebody slap me! :-D
Don't get me wrong, I didn't mean to suggest that because men in the OT could have several wives and concubines that this means it's ok for today. In most modern wedding ceremonies, especially Christian wedding ceremonies, there is a vow of monogamy made as part of that ceremony and for a Christian that vow is made before God as well. Breaking that vow is a sin! If someone wants to have an open marriage, they shouldn't take that vow!