In child custody, can a father have a live-in girlfriend?

My daughter’s father and I have had a verbal custody arrangement for the past ten years, mostly just a schedule & support arrangement, because we were never married, therefore did not have to go through formal divorce / child custody proceedings. However, his girlfriend is now basically living with him (sleeps over all the nights my daughter is there and many times in the same bed as her father). It’s not clear whether she even still has a place of her own to go to. If she does, she rarely stays there as she has now moved her workplace location close to his house. Nevertheless, this is not the example I want to set for my impressionable pre-teen daughter. I have spoken with her father about it, but he continues to do it.

Will creating a formal and legally-binding custody arrangement provide any rights that will guard against this behavior if / until
they are married?

What other benefits are related to having a formal custody arrangement?

A formal custody arrangement is beneficial in that your schedule is enforceable, and remedies exist in the event the other party does not comply with the agreement.

Creating a formal custody arrangement can bar the other party from having overnight guests of a romantic nature, but usually only if the other party agrees to such a restriction. Judges are hesitant to place such a restriction in a court order unless the third party romantic interest is shown to be a danger to the child.