How does a sponge keep its own filter from clogging?

Backwards, and out of the inlets. The sponge moves particle-trapping mucus against the direction of its own internal water flow and ejects it from the seawater inlet pores, the ostia, through periodic surface contractions that earlier work called sneezing. The result is a steady stream of mucus-wrapped particles leaving the animal, and a notable flux of detritus that sponge-associated fauna eat.

Sponges are among the oldest multicellular animals still with us, and they move an enormous volume of water. Seawater enters through thousands of small inlet pores, the ostia, gets filtered inside, and leaves through a few large openings called oscula. A filter that size has a clogging problem, and how sponges avoid it was largely unknown.

The natural assumption is that solid waste leaves with the outflowing water through the oscula, since that is where everything else goes. Kornder and colleagues filmed the Caribbean tube sponge Aplysina archeri in time-lapse and analyzed what it shed, and the footage shows the opposite.

Particles get trapped in mucus, and the mucus travels across the surface against the direction of the internal water flow. It gathers at the inlet pores and gets ejected there, during the periodic surface contractions earlier observers had already named sneezing. Watched at speed, the animal looks like it is continuously streaming debris off itself.

None of that detritus goes to waste. The flux coming off an abundant reef sponge is large enough to put a number on, and sponge-associated fauna actively consume it, which puts the cleaning mechanism inside the way a reef moves nutrients around. The authors note that mucus transport of this kind has close counterparts in other animals, humans included.