Hippocampal cells show precise spatial tuning to the animal's position during behavior. During stopping periods, the hippocampus encodes spatially coherent trajectories through the environment called replays that resemble the animal's behavior. Replay has been proposed as a form of virtual experience that supplements real experience for the purposes of learning and planning. However, it is unknown whether the precise spatial tuning of place cells as a function of animal position is preserved as a function of encoded replay position, independent of the animal's location. This is far from obvious, given that replay unfolds at much higher speeds and is likely to involve different mechanisms and circuits. We show that in a 2D environment, spatial tuning is preserved as a function of the encoded position during replay (its "replay field"), using a leave-one-out analysis that removes the cell's contribution to replay before evaluating its replay field. Replay fields are shown to match place fields on both coarse- and fine-grained measures, including spatial information, field shape and skewness, as well as in the relative positions and peak rates of the primary and secondary fields. Crucially, however, we show that spiking variability during replay is quenched (sub-Poisson) and thus does not exhibit the overdispersion or excess temporal variance that is characteristic of passes through place fields, suggesting that firing patterns during replay are less subject to covert shifts in attention and context as during behavior. Spiking reliability is typically only observed in sensory areas where stimulus sensitivities are well characterized, indicating that replay may be the equivalent of an idealized stimulus for the hippocampus. Lastly, we demonstrate that sub-Poisson spiking variability naturally emerges from a prominent, though largely untested, model of replay whereby spike frequency adaptation drives movement of a bump in a continuous attractor network.