If you don’t have consensus on who has which shares, you can’t possibly pay everyone accordingly. You can’t decide in a decentralized manner who to pay and how much. This is not a requirement that is reasonable to remove. You will end up inventing a consensus mechanism to do accounting of shares.
I will write more about the Braid mechanism as I’m porting it to Rust, but it’s very simple. Fundamentally it’s just Nakamoto consensus on a DAG, and there really isn’t much variability on how to do this correctly.