This looks like an acid burn on the slate, I had a similar job like this in Miami that no other contractor could fix, we went down there and used and alkaline cleaner scrubbed the slate and then used 250 grit black honing powders and then used the Color Enhancer form StoneCare Central and the job turned out great.
Recommend you test an area to see if you get the same results.
John E. Freitag
The Stone and Tile School
Can anyone ID this problem? Customer sent this to me and before I go running out to the house, I thought Id get some feedback.
Xtremeclean Tile - Boston Stone Restoration
Specialty Stone Care