But if I simply throw away my half and make a new half, a court will have no way of knowing which was the original. (Unless each half was signed by both parties.)
Your new half would not match together in the "split" part of the stick, so one could tell that /someone/ made a new half, but not necessarily which party was the forger.
Signing makes sense.
Both parties in a contract can have an incentive to lie.
I don’t think all contracts have an obvious both losing state though.
Think about two situations:
Firstly, if I lend you money, we split sticks, then I throw away my stick and get a new stick that says I lent you more money.
Secondly, if you lend me money, we split sticks, then I throw away my stick and get a new stick that says you lent me less money.
In both cases you're claiming my stick is clearly counterfeit because it doesn't match yours and I'm (falsely, but nobody knows that except the other party) claiming your stick is clearly counterfeit because it doesn't match mine.
What happens next? We're no better off than if we only had a verbal record and were disagreeing. So what is the point in the sticks?
Understand?
If we use a split stick to record how much money you owe me, and I simply add more marks to mine, to pretend you owe me more, it's obvious that I'm cheating. But if I simply split a brand-new stick and add marks on it, then our sticks won't match up at all. How will a court know which is the original?
Presumably they had some way to sign them? After all, that's why both parties sign both contracts today. If you go change yours, I still have your signature on mine.