I see what you mean. It does say built-in stabilizer on the box, but it is not a stabilizer per say. Homage has a setting which is recommended for Pakistan and it is called a "wide mode". In this mode Homage works between 90-280 V. This means even if the input voltage varies, as long as it remains between 90-280, Homage works fine.
This is important because in Pakistan sometime there are fluctuation (jhatka) on the line. And since Homage is designed to take over when it thinks the power has gone out, so the greater voltage variations gives homage enough time to decide whether it is a real power breakdown or just a false alarm.
This should work both when homage runs on mains and battery. It is at the input of the inverter not the output. So, it is not a stabilizer per say nor is it a AVR.
But in my opinion it is more important then a stabilizer for us Pakistanis as our distribution system is lousy.
I saw your email today. You should have mentioned that you have sent me a PM. That is my junk box and I only check it once a week. I have sent you an email back.