Brita filters might actually work. The outside surface of the pool could be studded with around 9000 screw threads onto which screw brita filters. That's one

Brita filters might actually work. The outside surface of the pool could be studded with around 9000 screw threads onto which screw brita filters. That's one filter per square foot. Now we just need to pump some water out of the pool to lower the water level inside and water will flow through the filters from the dirty side to the clean side. The pumping could be done the conventional way with electric motors, or you could rig up something clever with one way valves that lets the wave action do the pumping. You would need divers to replace the filters on a regular basis, but with a low enough flow per filter they could last a long time. The main problem would be biofouling of the filters, but a clever filter design might actually use the biofilms as part of the filtration medium.