Inverter outlet -> Power Source intended for Charger -> ???

How does power get to AC outlets on boat from the Power Source intended charger?

Or put another way, what breakers do you have on the AC panel?

I don't have a picture of exactly what is on each side of the problem GFCI breaker.

Sounds like you most likely must troubleshoot what is on the far side of the applied inverter power first.

I tell you, this is just plain un-netchurel !!!

So the recessed European shore power plug is male. Is it live when the inverter is active?

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All of this reminds me of the un-netchurel act I committed legitimately on my multi-outlet home phone system recently. I disconnected from Ma Bell's network interface in the basement in the vicinity of my punchdown block. Then I plugged in the magic Vonage box into my cable modem for voice over IP. Then connected my router downstream of that. The un-netchurel part was connecting the phone output of the Vonage box into the phone OUTLET in my office. Of course the phone signal propagates fine to all the other outlets in the house and phone wires are dual male (but low voltage!). Still seems un-netchurel!!! Just sign up for Vonage and you can get me a free month {grin}. I have been using it for a month at $15 a month introductory and it is feature rich. Phone service so far is not quite as good as Ma Bell.