I have done quite a bit of tile work over the years and aesthetically speaking, I do not think that the colors will "clash" against each other -- what you might end up with is something more along the lines of what your guests may view as "interesting" and you may genuinely like. As a contractor, my recommendation would be to use the white grout on the walls because it will stand out nicely against the white and gray floor AND it will look cleaner for a longer period of time (as grout ages --- say in about 10 years time if it is set properly and sealed with grout sealer -- the grout may crack in some corners or other wetter places in the bathrooms at which time any grout no matter what the color may start to discolor, fleck out and may be in need of repair to the grout (be sure to ask your contractor to seal the tiles with a tile sealer for max use and wear and tear -- I use Tile Lab Grout and Tile Sealer which you can get at Home Depot or Lowes, but your contractor may use another product that is just as good because there are several brands available).
I do not think it will clash -- it will blend with the floors nicely. If you were doing some really strange color then I might have some concerns regarding how the finished product would look -- but greys and whites and tans are very common grout colors and I have seen them all intermixed in the same bathrooms and the same kitchens for the past thirty years.
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