looking for a perennial fruit bush that will grow on latice

Hello

I just hung some shade cloth for my meat rabbits as the only available space in on the west side of the house for them. I am looking for a perennial fruit bush that will climb lattice and keep its leafs during the summer.
I am in USDA zone 9B. In the mean time I going to be planting sunflowers again to provide the rabbits more shade. I was young in this photo, I am 6 feet 5 inches tall.

I'm not sure if this is for sun or shade.
If it's for shade maybe it will grow on the shady side of a lattice panel.

I am thinking grapes, maypop-passionfuit, kiwi, etc. Maybe thornless blackberry will work too.

Seconding grapes for this. Traditional for shade structures worldwide, plus you get fruit.

I think rabbits will eat grape leaves as well?