Catholics do not recognize poly relationships and Catholicism has never condoned poly marriage, or spouses having multiple partners. If the people involved are all not married, then having sex is fornication, regardless of whether a person is having sex with one person, two people or 200 people. If it is a case of a married couple and one spouse gets romantically involved with someone outside the marriage, then that spouse is committing adultery, plain and simple.
Many people would say that a spouse should not have to tolerate the other spouse being openly adulterous. I understand however that sometimes the non-cheating spouse will want to try to hang in there and save the marriage, or there are children to consider. I recommend you talk to your priest.
Even if you have somehow decided that for the sake of the children, you are just going to put up with his behavior and turn a blind eye to it, you should not have to share your marital home with his new girlfriend. Not only is that grossly unfair and hurtful to you, but it would set a terrible example for the kids. Your spouse is probably thinking that since he’s gotten you to more or less agree to letting him have a girlfriend, he can push a little harder and you’ll agree to let her move into the house. You need to put your foot down and say NO or I am afraid of what your spouse will expect you to agree to next - maybe that you move out and he shares the house with the new girl and the kids.
I have known non-Catholic (and for the most part non-religious) people who were in actual poly relationships (without kids). Three people living together generally does not work. Someone will always end up feeling like they are not getting enough attention, or feeling jealous, and the triads break up. The only time triad relationships might work is if one is living in some culture that isn’t normal Western culture, where multiple partners - usually a man having several wives, or one wife and several concubines - are tolerated. Even then it often creates a very bad and abusive situation for the multiple wives, who often do not like the arrangement but don’t have enough economic or societal power to leave.