No, it could be rated for 180VA.
It might have 36 turns on the primary and 324 turns on the secondary, for example, with a primary voltage rating of 12VAC and 180VA rating.
As this is not homework, I've provided a schematic below showing what I had in mind, but for a real application other things would come into play (such as the waveform at the inverter output).
simulate this circuit – Schematic created using CircuitLab
In this circuit the transformer boosts the 12VAC to 108VAC, and that is connected in series with the 12VAC input to give 120VAC, so the transformer only needs to be rated for 90% of the output VA, or 180VA in this case.
Note that it does not give isolation.. this is an autotransformer buck/boost configuration.
The input still has to provide 200W, of course, so 16.8A plus transformer losses.