Essentially the question is what do you do with the excess solar power. Without batteries, the excess power is just shipped out to the grid. But if there is no operational grid, the solar system will shut down the inverters since if send the excess power out to the grid it could shock workers trying to restore the grid.
But the Powerwall Gateway also monitors the battery and grid state. If the batteries are full it uses the batteries to power the house and after the batteries drain down a bit brings the solar back online to replenish the batteries, and takes solar offline when the batteries are full. The gateway can repeat this filling and powering from the battery from solar cycle for a very long time without the grid. Some members here, like
@MorrisonHiker have purposely gone over a week being disconnected from the grid.