I've had horrible luck finding a product that works in this situation. What are you using currently for man made tiles?
I've tried lots of stuff. Some, like Viper Renew, are powerful and have only the drawback that they can rust stainless, or make blueish stains on chrome. However they are fantastic for softening efflorescence and hard water deposits on tile, as well as leftover grout on the tile face.
For general purposes, I make a slurry of Barkeeper's Friend (powdered oxalic acid) and warm water. Dab it up on a piece of red pad pad cut to a 5" or 7" disk and scour the walls, chrome, glass and floor with it on your Makita. The 5" drive disk is available at Harbor Freight for $7. Spritz a bit of water and scour away. It also works well with a stiff nylon brush.
You can run it right against the cruddy edges and it will save you a LOT of labor and aggravation.