does chasis battery charge while generator is on?

Our coach charges both with shore power and the genset, but only after the house batteries are fully charged, as they get first priority. We have only the converter/charger for this and the inverter is a separate after market unit, that I installed myself.
I leave the coach plugged in at all times anymore and it's never dead and doesn't boil the water out of the batteries either.
The engine alternator charges both banks at once.