The two bathroom basins in my 1986 house (finished by a handyman-occupier) are different. Downstairs there is a rubber (?) gasket between the basin and the tile surface into which it sits. Upstairs there is no gasket (or caulking) -- so moisture creeps round the edge of the basin at one point, then drips down underneath.
What can I do to fix this leak?
Remove basin, caulk correctly. If you want to stop the problem right away, but need to put off the reinstallation for a while (like, until fishing season is over), you could run a bead of caulk around the edge, but it's going to look awful, and get in the way when you need to wipe down the area.
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