Not sure about why the sync issue is caused by a lack of speakers, but I get the same issue when accessing my HTPC via TeamViewer and the Amp is off, suprisingly it always happens when I'm not at home
If you're passing sound via the HDMI to the AMP, when it's off, Windows assumes it's had the speakers unplugged or the soundcard is disabled. With no sound card available, some things don't work in MB or other apps, related to sound. So, my fix has been that I happen to have an internal soundcard on the mainboard, which it can revert to when the AMP is off. Stick a 3.5mm jack plug into the speaker socket, so it thinks it has speakers, and it will fool Windows into thinking it has a sound card and speakers attached.
When I connect to the HTPC remotely I goto playback devices and make the internal soundcard the default before loading MB.
See if this stops the sync issues, if they only happen remotely?