Does the DC portion of an inverter experience the Apparent Power consumption of the AC part?
The dc section of an inverter has to provide power even when there is no load on the ac output. Nothing is 100% efficient and the off load dc power consumption fuels the losses in the switching transistors and output filter. These are real and not apparent powers.

Should you connect an ac load that takes zero power such as a pure reactive component then the dc losses increase a little because more current is demanded from the ac output.

This increased dc loss would be approximately the same should the ac load take the same current but be resistive rather than purely reactive.