With how our inverter is set up it is also possible to have it switched on with the shore power plugged in. But it is impossible for both the shore power and the inverter to provide power at the same time. The shore power lead is plugged into whichever source you want power from. So it can't be in both at once.
That's fine but not what JonathanA was suggesting.
Note there seems no point in having an inverter running and drawing standby power from batteries when its (correctly) not possible to use its output due to presence of shore power. The advice to turn it off stands therefore.