If two TCAS equipped aircraft are in an encounter the two systems should both issue and RA (and they are coordinated). They won’t necessarily be corrective RAs, one or both could be preventative.
A few other considerations.
1. At the risk of stating the obvious, did you possibly have TA ONLY selected? (Finger trouble on line up)
2. Could there have been another aircraft involved? I.e. giving one another an RA while you were a bystander?
3. Possibly a non TCAS light aircraft on a VFR squawk suppressed by ATC but blundered into CAS? (This has happened historically)
3. TCAS has different sensitivities by altitude, perhaps you were both either side of a boundary? My understanding is that regardless if Either box triggers an RA it attempts to initiate the coordination data linking., but maybe I’m wrong.