Filtering can remove some of the beneficial components of kombucha. How much filtration are you looking for?
A simple strainer will work just fine for the yeast strands and any chunks of material.
If you want tea sediment too, you could use a nylon mesh, like a reusable coffee filter or regular paper filters. Paper filters are by no means fast.
Or you could put the liquid in a sealed container capable of holding pressure, fill it almost all the way, ~1" from the top, and let it age in a cool dark place for a week. This will switch the bacteria to anaerobic, dry out a little more of the residual sugars, and allow the unwanted mass to collect at the bottom. You could rack from this secondary vessel, or add priming sugar to carbonate too.
But so long as there is live yeast, there will be sediment. You'll never get it all without fancy mechanical filtration.
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