A UPS pre-sync's to AC input sinewave zero crossing so it is able to bring up its inverter in sync with what was the AC input phase before it dropped. When grid come back on the UPS keeps running from battery for a while and gradually resyncs its inverter to grid before turning output back over to grid. Proper grid voltage range is also checked before turning over the load to AC input.
Problem with modified sinewave for UPS input is it does not provide a succinct zero crossing point so UPS does not have a sharply defined zero crossing to follow. Its sync process jumps around telling the UPS controller it has not achieved proper AC input synchronization so it will not take over AC output load.
Couple of other incidentals are AC voltage measurement/matching and possible power factor correcting charger in UPS.
AC voltage measurement is likely just a scaled AC sinewave full wave rectified average, like a cheap DVM. Not a true rms AC voltage measurement. Modified sinewave will read incorrect rms voltage equivalent on an AC average responding meter so UPS AC output voltage matching will be off. Not a showstopper in most cases but the UPS inverter will supply a slightly lower AC output voltage as the result of inaccurate AC rms input measurement of an input AC modified sinewave rms voltage.
Especially in Europe with its power factor regulations, the UPS battery charger might have a power factor correcting frontend. PF correction circuits expect a sinewave input. Some can be damaged if input is modified sinewave input or will shut down if too far off from a true sinewave due to high peak current in PF correction circuitry.
Many computer power supplies these days have power factor correction. This is why most new computer UPS supplies are now sinewave output. Old UPS's often had modified sinewave inverters to supply backup AC output. The old UPS's still required sinewave input to acquire proper sync.
