Sediment

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That was exactly my approach
One catch - siphoning into another container seems to completely remove the bubbles, so don't do it if you want sparkling wine.
All of my attempts so far have remained cloudy, however After two days in the fridge I just go ahead and drink them anyway.

Question: how many days does it take for, say, grape wine or cider to completely clarify?

If I want really clear (which is rare, usually only with beer) I usually let it sit until its mostly clear (3 days max) then I rack it. Its usually clear the next day.