I think the issue is is the terminology "hybrid inverter". Are you thinking of a grid tie inverter like this one?
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This is an example of an inverter that will shut off when the grid will go down.
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It has what is called "Islanding Protection", which is what you are talking about. It has three inputs/outputs. ACout for off-grid power. It also has two ACin sets of terminals for grid and generator input. Islanding will work if the grid is connected to ACin. It will NOT work if the grid is connected to ACout.
If you want to connect the grid directly to ACout, you MUST use the transfer switch/toggle breaker described above.