I know the CLUSTER command essentially creates a new table, copies the data in INDEX order, then deletes the original table.
Does that mean all of the data passes through the WAL logs, or is it purely an internal operation? More specifically, if we're doing streaming replication, will all of the data for the table being clustered be transmitted across the network to the hot-standby slave?
Not looking at the code but, yes, the replica must be an exact physical duplicate of the master and since the only ongoing source of data for the replica is the WAL the actions taken as part of performing a CLUSTER must be logged.