Oh... good ideas.
definitely not for the desert islands unless they have a compact way of doing it.
It could work even better offshore — a sort of cell structure, like a beehive, brine would sink, heat would exchange, fresher water would rise, sun’s rays would evaporate it, and the part of this floating flexi plastic that rode high would be continuously evacuated by blowers, carrying the hot vapor-laden air to shore, up one or more black rigid tubes getting that last bit of solar heat, then plunge through the tank wall, across the water that was already collected, there to condense. Simpler, maybe even cheaper, probably easier to operate, and unlimited size. Also would have the virtue of not consuming the shoreline, which would then be (trickle-) irrigated for agriculture as well as general refoliation.