Is this a relationship or a court case? If you’re waiting for the words to fall out of each other’s mouths just so you can say “Aha! Caught!”, then brother, you both have considered each other not partners, but enemies.
You say you’re tired, but the truth is that as long as you both have the habit of “winning”, this fatigue will never end. There is no “winner” in a relationship; either both win or both lose. Right now, you’re both prepared to lose.
Frank, if you want to save the relationship, you’re going to have to be a little “big guy”. Stop reacting blindly and start learning how actually to respond. And if both of you are enjoying fighting, then rent a boxing ring, at least there are rules there!
AskApril, what’s your take?