The thing that is not clear is this: Why does your girlfriend get a say in whom you date? That is downright strange.
Is it because any new prospects will have to be involved with your current girlfriend, too?
That is what BU meant by "are you looking for a third?" In other words, a new sex partner for the both of you. And by the way, there was nothing disrespectful in how she put it. You chose to take offense, but didn't have to. We are only offering our viewpoints, no need to get all huffy about responses you don't like -- take them or leave them.
Or is your girlfriend just that much of a control freak that she demands to have approval (veto power) over YOUR relationships? That is usually considered quite a bossy, oppressive policy, in the view of most polyfolk.
If either of those is the way you are trying to do poly, no wonder it hasn't been successful.
Why shouldn't each of you date separately? Why does your girlfriend expect to have a say over who you want to be with? Is she your boss or your partner/equal? Why doesn't she trust you enough to make your own choices? The only time it could possibly be appropriate to meddle in your relationships would be if there is some danger to warn you about.
It does sound like she really is not on board with poly and, therefore, not actually giving her consent. I recommend much more talking and negotiating, and perhaps even therapy or counseling, to strengthen your foundation and get on better, solid ground before pursuing other relationships.