My wife always says "Check the connections". This is because often times she hears me say, "^&^&%$$% Connection" in regard to electrical
gremlins. Make sure they are all clean and corrosion
free. I'd also be concerned about the gauge of the wiring. Not just to the compressor itself, but to the breaker panel too. More emphasis on the size of the wire to the compressor. Most of the modern refrigeration
units have under voltage protection. They will shut down when they detect a low voltage threshold. This keeps them from completely draining your batteries if nothing else.
Since your unit is working when connected, I'd think you reaching this low voltage threshold. After check all connections, you should measure how many volts are available at the compressor when you're both connected and disconnected from shore power
. If there is significant difference, you will need to find out the source of the difference. Is it batteries? other drains? etc...
75 watts of solar
is probably not enough to supply the complete load of your boat's systems. You'll typically see that output only at ideal conditions. Optimal sun, optimal battery charge state, etc. So, where is the 13.8 voltage being measured? Are there no other loads on the system?