OMG - I think I've written this same type of post before LOL! I was just talking about this with a friend of mine, both of us are those "mean" parents, just like you.
I actually think that the main issues are 1) respect for other people's homes, 2) different parenting styles and 3) expectations for disciplining other people's children.
Overall, I think that bottom line, in your home, make your rules and enforce them equally for ALL PEOPLE (not just children) that come into your home. After that, I don't think it's your business to control her kids.
I agree with your 100% that allowing children to be out of control in any situation is unacceptable to me too. However I have had to learn to respect other people's parenting styles and expectations of their own kids.
If you want to, try to talk to your girlfriend. Explain that your are simply trying to "help" keep all the kids safe and at a reasonable volume. Don't blame her, or her children for their behavior or lack of control. That's YOUR prerogative. Explain that you don't want anyone getting hurt, anything getting broken, etc. And that you hope she agrees that when things get "out of control" with ANY of the kids, that each of you should stop the given behavior. THEN, be quiet. Let her think, answer, vent, whatever. Just give her a chance to voice HER opinions.
I think bottom line is that if you don't agree with her children's behavior, she won't do anything about it and doesn't want you to either, then you have to decide if it's worth having them in your home/backyard? Unfortunately, there are cases where I have decided that since some people CAN'T respect my home, my rules, etc. then I can't have them over. Instead, we meet at the zoo, park, etc. And when their kids start acting up, I either ignore it or start walking away with my kids and refocus them on something truly interesting.
Msg me anytime. It sounds like you and I have very similar views on parenting.
I know it's frustrating. But I think you have to just let her deal with her kids.