There is a good chance that ATC will become urgently concerned during a TA or RA.
Both a/c follow RA -- OK
Both a/c follow updated ATC instructions -- OK
One a/c follows RA and other follows ATC
in accordance with RA -- OK
One a/c follows RA and other follows ATC
against RA --
Once RAs are issued, it seems there is an even probability ATC's altitude solution will be opposite to the one from TCAS.
TCAS has been designed with two players in mind -- the crews. But ATC is there too and possible ATC interventions need to be accounted for.