I don't know anything more about Spanner than was in the paper, but my impression is that Spanner is doing normal two-phase commits and the paxos-elected leader assigns unique timestamps.

The unique thing seems to be that when you want a read-only query, the client will ask for latest safe timestamp and then it will ask the nearest replica what the data did look like at that specific point in time.

However if you want a read-modify-write transaction the client will need to talk specifically to the master replica and lock the data, just like in a normal db.