I'm guessing that my charger/ inverter is the culprit? It is a digital unit and is showing 13.4 volts out with AC power in while charging.
I'd follow through on CD's question if I were you. The same applies if your boat has a wind generator or other second source. It need not be your inverter or the other unit per se, but the circuit.
Remember that current flows where there is a voltage drop, and you may have more than one "drop" in the system if several sources run simultaneously. It is possible for the current to push from
the shore power circuit into the solar/windmill system when the voltage of the latter is lower, with some unexpected results. There needs to be separate voltage regulators on each device, and diodes blocking the current from taking wrong paths.
I had a related issue with only the engine generator and windmill running while at sea, pushing the voltage up to 16V in the worst case. In this case the engine generator was the main fault, but at a boat show I cornered a Finnish superguru who went gaga at the question; he'd been waiting to launch into a half-hour lecture on the intricacies of such circuits.