Nothing happens, the controller will open the circuit and prevent the current from flowing.
Optimal solution is biting the bullet and getting batteries, the controller will prevent them from being overcharged, and you will need stored energy on an island.
You could try gravity storage like the other person mentioned but it's less efficient. During the day the solar panels power a pump/motor to move water/heavy object up high, and at night the water/object moves back down and powers a generator. There's a lot more loss since you're going:
solar energy>electrical energy>rotational energy>hydraulic energy/potential energy>rotational energy>electrical energy
and each transition will have losses due to friction and heat.