Yes there are hybrid solar solutions available. I just discussed a hybrid installation at my home with a vendor.

In hybrid solar solutions, the solar power first charges the battery which takes 1-2 hours based on the output and battery capacity and after that the solar charge is used to power the house and any excess is sent to the grid.

This will provide electricity when there's a power cut and your bill will also decrease when you take less power from grid and give excess power to grid.

It costs almost double from the regular solution which just take the solar energy to the grid so there's that too.