Stationary coalescing walks on the lattice

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.