Context
Suppose I program a blockchain which doesn't require proof-of-work and deploy it on 3 servers A, B and C.
Now suppose server A and B receive requests to insert a new block with some data in the blockchain. Both A and B modify their local copies of blockchain. Before the modifications, the local copies of the blockchain are in sync. After the modification block a1 gets inserted in the copy of blockchain A, and block b1 gets inserted in the copy of blockchain B.
Question
How do I sync the blockchains in A and B, and what will the blockchain on C look like after a successful sync?
The previous block hashes on a1 and b1 will be same. However, after a sync, which one goes first? which block's previous hash get's modified? And most importantly, is there any established protocol to accomplish this synchronisation?
I hope my question was clear, can update the question with more details based on comments and answers. Thanks.