  Coalescing simple random walks in the plane form an infinite tree. A natural
directed distance on this tree is given by the number of jumps between branches
when one is only allowed to move in one direction. The Brownian web distance is
the scale-invariant limit of this directed metric. It is integer-valued and has
scaling exponents 0:1:2 as compared to 1:2:3 in the KPZ world. However, we show
that the shear limit of the Brownian web distance is still given by the Airy
process. We conjecture that our limit theorem can be extended to the full
directed landscape.
