In Discovery, consensus happened on Ethereum. Each worker was also essentially an Ethereum light client. Transactions were ordered on Ethereum through Task Records, and finalized through their accompanying task receipts. Workers in Enigma, as mentioned, replicated the data completely to avoid data availability problems, but they would only do so to mirror the task records/receipts on Ethereum.
That said, we have since moved on to a new architecture where there is direct consensus (i.e., the protocol itself is now a blockchain, and not a second layer network).