OK! The solution to the problem was to order some oxygen bleach (it was cheap), mix it up in a bucket of hot water, spread it out on the filthy tile and let it sit for an hour, then hit with a deck brush. Viole`. Looks like the day they laid the tile.
Easy as can be.
Now I tried TSP, 409, scrubsoap, etc... scrubbing on hands and knees with brillo pads for hours and got almost nowhere.
The Oxygen bleach was like MAGIC!!!
I did read there is some special bathroom tile that nothing works on except grinding and sealing but that aside, this stuff cleans any tile I put on it making it look brand new.