Thank God! I've been thinking I was crazy about this for several years now!
My memory of the last time I tried to concisely figger all this is that whenever there is AC, the batts get charged. (and, I guess when the alternator of engine is running, as well)
I cannot say for sure anymore that my bstts are linkrd in the absence of any other power source -- I was pretty sure of this at one time, but haven't tried to verify once I put a terminal disconnect on the house batt so I could physically ensure that the two were indeed separate (I rarely bother to use this disconnect anymore, and see no issues with the batts with RV in STORE between monthly runs)