> isn't it complicated to have their output be in perfect sync with the frequency

Not especially, given that the inverter has a microprocessor in it. All it has to do is measure the phase of the existing grid.

I don't have references for how it's actually done, but one obvious approach is simply to wait at each zero-crossing for a new half-cycle to cross a voltage threshold before turning on the output. This also implements the requirement to drop out if the grid goes away. It is probably also possible to measure during the "off" side of inverter output PWM, in the same way that variable frequency motor drivers work.