Our lanai faces West also. We planted 3 trees ( 2 loquat that had been shaped at the nursery to be canopy and a lugustrum). Probably spelled that wrong.... anyway we planted them (lugustrum in the center) about 7 feet apart and they have grown to provide full shade into the lanai and the ceiling fan is on most of the time. We use our lanai alot and most of the time we are comfortable out there. Good luck.... hope I've given you something else to think about as a possible solution.
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