South of the Snow Hill tunnel (I'm not sure where exactly; I always thought it was City Thameslink but based on what has been said on here - and not just by Robinson above - it seems to be Blackfriars), the FC+SE services are run by SE. At no stage are they considered a truly joint operation; it's merely a case of Operator A running the service up to a certain point, and then Operator B takes over.
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Clue 1: This station has some services running through non-stop during the peaks.