It is definitely water trickling down. If you take a look at the
branches going horizontally to the left, you will notice that the
discoloration is on the top of the branch also. What beats me is why
not on the sides. The tree stem and branches seem to be having plenty
of nodes and the discoloration seems to be most along the nodes.
Possibly the bark is more porous along the nodes as such absorbs more
moisture and hence more soot or dirt sticking along the nodes.