What would be a good option for a non-slip entryway surface in an anteroom where the door is very nearly flush with the floor?
One option may be to use a non-slip paint additive mixed with an epoxy or compatible tile sealant. I've used these on exterior pre-cast concrete stairs before with great results. There are also several brands that are specifically marketed for pool decks that are part of a "coating system" that is designed to be used over tile.  If you have a local pool supplier, they may be a good place to start.

Note - since the tile is already laid, I'd either use a spare (if you have access to one) to test the adhesion or trial it in an inconspicuous place before you go forward with the whole thing.