Try this test.
1 Disconnect shore power.
2 Get a volt meter.
3 Turn on your inverter.
4 Test each leg on the shore power cord. Be careful as they might be hot if you are in fact back feeding the RV distribution panel.
If the prongs on the power cord are hot while the inverter charger is on you have a very dangerous set up.
Also if you're running the AC unit off the inverter/ batteries you're probably boiling the batteries. Another scary application.
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I am almost positive is cant back feed since the Progressive industries contactor is off when no shore power is detected and I do not have any power on L2 when on inverter power. When the AIMS senses no input AC it will supply AC to L1. The load created from the air conditioner shuts the inverter down so I am not worried about the batteries. They are also protected via the 300A fuse. Battery-300A fuse-cut off switch-inverter. I also have a catastrophic fuse on the positive side of the battery bank.