  We consider translation invariant measures on families of nearest-neighbor
semi-infinite walks on the integer lattice. We assume that once walks meet,
they coalesce. In $2d$, we classify the collective behavior of these walks
under mild assumptions: they either coalesce almost surely or form bi-infinite
trajectories. Bi-infinite trajectories form measure-preserving dynamical
systems, have a common asymptotic direction in $2d$, and possess other nice
properties. We use our theory to classify the behavior of compatible families
of semi-infinite geodesics in stationary first- and last-passage percolation.
We also partially answer a question raised by C. Hoffman about the limiting
empirical measure of weights seen by geodesics. We construct several examples:
our main example is a standard first-passage percolation model where geodesics
coalesce almost surely, but have no asymptotic direction or average weight.
