I take it we are talking about a SMB built rig?
The question is a little hard to answer with out know more details, but first we will give you some general theory.
When plugged in to shore power the Tripp-Lite Inverter/Charger will act as a charger for the house battery system, as long as it is set in charger mode or Auto.
When driving the alternator will charge the starting battery bank . This could be one maybe two with diesel start batteries. There are some configurations that had a factory aux battery, but we will leave that out for now.
Now you ask the question on how does one charge the house battery when driving via the alternator, and how does one charge the starting battery or batteries when plugged in. It is a reasonable question and I wish I had a simple answer.
This is generally accomplished by a separator, isolator, combiner, ACR type device. They all pretty much function the same except if you have a diode based isolator, it is slightly different then a relay based isolator. These generally work on the premise that if there is a charge voltage present, they will combine the two battery banks so that either charging device can charge all the batteries.
For a 2006 SMB the common device was a Surepower 1315-200.
So you really need to go look and see if you can find one of these hooked up. If you have the original I would expect it might be giving you some issue at this date.
It could have been replaced and normally that is with a Blue Sea ACR 7620 or 7622
Depending on who installed it, they may or may not of used the switch seen in the picture,
So go look and see if you can find one of these units, if it the SurePower it was pretty much all automatic. The BlueSea ACR can have a switch that would provide a off, auto,on switch. If you have the BlueSea ACR with the yellow top button, it would give you a manual settings.
Once we know which device you have, and we can go from there on the details of how that one works.
-greg