Beneath my window is a path which leads up to a private dwelling. Part of this path is covered by a tiled roof. Would I be able to do mayim achronim from my window onto this tiled roof.
In principle, mayim acharonim should be done over a keli, and should not be done over the ground. However, if the water will not fall from the tiled roof onto the ground, one may be lenient, for the roof is not considered as the ground. Nevertheless, it is preferable to pour over a vessel in which the water is collected.
Sources: Chulin 5a; Rambam (Berachos chap. 6) is stringent, and writes that mayim acharonim should be done specifically over a keli. Shulchan Aruch (181:2) rules that one should preferably use a keli, but if this is not possible, one should pour over something that separates the water from the ground (see Magen Avraham 181:2). This would apply to the tiled roof.