Mining & The Double Spending Dilemma
You just have to wait. Eventually, another miner will produce a block and it will be on top of one or the other of the two blocks found at about the same time. The probability of again having two miners find blocks at about the same time is very low. The truly paranoid can wait for six blocks to be mined before relying on a transaction.

Persistent splits should be unlikely because the longer the chain you mine on top of, the higher the probability of you getting to keep the mining reward. If you mine on top of a chain that is one block shorter, you only tie the existing chain that other people are already mining on top of.