The longest chain with integrity wins, miners that detect a longer chain with a double spend do not have to give up a join the longer chain. They can very happily keep mining on the shorter chain they are on. Effectively become blind to any chain with integrity issues. Part of giving up and switching to a longer chain is a simple integrity check on that chain.
All distributed systems consensus base have to handle split brain.
Implication: It may take more than 6 confirmations to be considered irreversable. But not much more…