The common perception is that strong coupling to the environment will always render the evolution of the system density matrix quasiclassical (in fact, diffusive) in the long time limit. We present here a counterexample, in which a particle makes quantum transitions between the sites of a d-dimensional hypercubic lattice while strongly coupled to a bath of two-level systems that “record” the transitions. The long-time evolution of an initial wave packet is found to be most unusual: the mean square displacement of the particle density matrix shows long-range ballistic behavior, with ⟨n2⟩∼t2, but simultaneously a kind of weakly localized behavior near the origin. This result may have important implications for the design of quantum computing algorithms, since it describes a class of quantum walks.