Actions that take place at the same time. e.g. I try to move around the tank to hit the glass cannon, while the tank tries to stop me. There are many RPGs that do this.
That said, since I was suggesting that the problem was turn-based initiative, I'm open to arguments that it's not the problem.
Can you name a few? How long do their combats typically take? Are they gritty 'easy to die' systems or are they 'hp and multiple-round combat' systems?
I've literally never seen it done well.