Dull the surface of the old tile using your vibrating sander and fine aluminum-oxide sandpaper. Wipe off the dust.
Apply polyurethane onto the tiles with a paint brush, in a thin, light layer. Brush it only onto the tile surface, not the grout between the tiles. Make the brush strokes go only in one direction.
Let the polyurethane set for 12 hours.
Hand-sand the polyurethane with extra-fine regular sandpaper, just enough to dull the gloss. Wipe off the dust.
Brush on a second layer of polyurethane. Let it dry for 12 hours.
Hand-sand the polyurethane as before, dulling the shine and wiping off the dust. Wipe up the dust.
Brush on a third layer of polyurethane. Allow it to set 24 hours.