  We prove a quenched invariance principle for a class of random walks in
random environment on $\mathbb{Z}^d$, where the walker alters its own
environment. The environment consists of an outgoing edge from each vertex. The
walker updates the edge $e$ at its current location to a new random edge $e'$
(whose law depends on $e$) and then steps to the other endpoint of $e'$. We
show that a native environment for these walks (i.e., an environment that is
stationary in time from the perspective of the walker) consists of the wired
uniform spanning forest oriented toward the walker, plus an independent
outgoing edge from the walker.
