You want any of your charge sources to be charging
both batteries as needed. If you spend any significant time motoring and away from the marinas
, then certainly you want your alternator
to charge both house and starting batteries. If you are at anchor
you want your solar to maintain your starting battery and charge your house battery. If you leave the boat in a marina connected to dock
power for weeks, then you want your house charger
to maintain both your starting and house batteries.
For this you may get an FET battery combiner and connect all 3 charge sources to the input terminal. Each battery to it's own output terminal. This subject can be controversial but I like Victron's FET combiners. Very low voltage drop with these: