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The problem is how to keep such a system clean, as then there will be a bunch of foreign gunk floating around in your cistern.

Yeah and bacteria would build up in the reserve tank.

An inline filter of some sort could be used for particles but that would require maintenance.

There would also be soap scum build up and hair conditioner products would gunk up the works I'm sure.

In Japan they don't seem to have a problem to put a sink (for handwashing) that drains to the reserve tank: