My wife and I took an OeBB train from Innsbruck to Verona last October, which coincided with an Italian rail strike. Fortunately for us, the OeBB-operated train from Munich ran (on time too) though it was terminated at Verona and those going on to its normal terminus of Bologna were bussed from Verona. Can't remember if the SüdTirolean trains were running.

It'd seem OeBB (and DB?) doesn't trust the Italians as it has an Italian subsidiary operating its own locomotives. Additionally, OeBB and DB at least have a joint ticket office in Verona which was open while the TrenItalia rail ticket office was closed.

Not sure how the other OeBB/DB joint-operated trains were running that day.