Monogamy is imperative when the relationship started that way and 1 person still wants it. It's a hill worth dying on. You are well within your rights to basically say, "if you pursue a romantic or sexual relationship, emotionally or physically with anyone else, I will consider you to be a cheater and this relationship will be over!"

There are "ethical non-monogamous" relationships (like polyamory) out there but those only work when everyone wants it, everyone will benefit emotionally and physically and in all other ways from it, otherwise this is a nightmare in the making. If you cannot see yourself pursuing sexual or romantic relationships with other women, while in a relationship with your gf, and be supportive and happy with your gf while she dates and fucks other guys, then you should absolutely die on this hill.

Do not go there. You clearly don't want it.