Not so much the phase relationship, per se, but there's always some possibility that the phase difference manifests itself as a ground potential difference in the low voltage circuitry, particularly if the low voltage ground is somehow tied to the AC neutral.
I don't know about your specific combination, but I've had different logic circuits interconnected with differential transceivers and being supplied from different DC supplies run from different AC mains. On power-on, the difference in AC phasing wound up coupled into the digital grounds and the net potential difference was equalized through the transceivers. They lasted on average about 3 power on/off cycles before the emitter juntions in the receiver inputs died.
If the equipment are intercoonected, they'll need to have a continuous ground connection between them to prevent any untoward behavior.
TTFN