Pour distilled water on the rusted areas to fill in the pores of the slate tile.
Create a poultice of diatomaceous earth and rust remover in a bowl. Add enough of the ingredients to make a thick paste. Hardware stores sell both products.
Apply the poultice to the tile with a plastic scraper. Make the paste 1/4 to 1 inch thick.
Place a small sheet of plastic food wrap over the poultice. Tape down the edges of the plastic wrap.
Let it sit 24 to 48 hours. As the poultice dries, it absorbs the rust stain.
Remove the plastic wrap and scrape off the poultice with the plastic scraper.
Rinse the slate tile with a sponge filled with distilled water.
Dry the slate tiles completely with a clean cloth.