What is the exact consensus protocol Ripple uses? Ripple supposedly incorporates transactions into the ledger using a consensus protocol. I have been looking everywhere for a clear, formal, and precise specification of the protocol and could not find one. 

Several related questions, e.g., 

What are the pros and cons of Ripple&#39;s consensus as compared with Bitcoin&#39;s proof-of-work?

and

How does Ripple solve the double-spend problem?

Only provide a vague description. More specifically, I am trying to understand the essence of the consensus: what happens if the Ripple network is split in half (for example if all communication lines between Europe and the US suddenly stop working), and each half of the network approves a ledger with a conflicting transaction. What happens when the network reconnects? How are the conflicting ledgers resolved?
If the Ripple network splits in half, then each half will validate ledgers independently, potentially with conflicting transactions. If all servers are properly configured, they will not accept any of these ledgers as fully validated, since none of them will have sufficient validations. It's possible both sides will think they're in the minority (if the split is close to 50/50).

This reflects a basic Ripple design principle -- don't tell people they can rely on results if the results are unreliable. If conditions make reliable operation impossible, it is preferable not to operate than to give people results they cannot rely on. Ripple is designed to detect these kinds of conditions.

When the network rejoins, servers will see the other ledger chain. They will gradually avalanche to the chain with more validations until one chain or the other has a super-majority of trusted validators. At that time, the network is back in agreement and everyone can again rely on the results of transactions.