Hi Neil
We see this problem on many tiled floors especially deep riven floor tiles.
This is usually caused by a combination of cotton string mops and high alkaline based floor cleaners.
Traditional cotton mops or as some people call them (CFU mops) just push the soils around the floor surface when the moisture evaporates the soils just dry back on the surface, this problem is compounded if the janitor is using a high alkaline based floor cleaner (caustic and corrosive ingredients) on a daily basis as they just grind the soils further into the grout lines,crevices,nooks and crannys etc on the tile surface.
Best and safest solution to this problem Neil is to pre spray the floor with a micelles based heavy duty degreaser (non-caustic,non-corrosive) diluted at 5-1 for deep cleaning, let dwell for 2 to 3 minutes then scrub with a grout brush or cylindrical scubbing machine for larger areas, this will remove all ingrained and compacted soils, use a wet pick up machine to vac up all the mess, then finally just flat mop with waffle microfiber floor cloths.
If i can be of any further assistance Neil, please just let me kinow
Kind regards
Tadgh