  There is widespread disagreement about how the general covariance of a theory
affects its quantization. Without a complete quantum theory of gravity, one can
examine quantum consequences of coordinate choices only in highly idealized
`toy' models. In this work, we extend our previous analysis of a
self-gravitating shell model [1], and demonstrate that coordinate freedom can
be retained in a reduced phase space description of the system. We first
consider a family of coordinate systems discussed by Martel and Poisson [2],
which have time coordinates that coincide with the proper times of ingoing and
outgoing geodesics (for concreteness, we only consider the former). Included in
this family are Painlev\'e-Gullstrand coordinates, related to a network of
infalling observers that are asymptotically at rest, and Eddington-Finkelstein
coordinates, related to a network of infalling observers that travel at the
speed of light. We then introduce "inflying" coordinates - a hybrid coordinate
system that allows the infalling observers to be arbitrarily boosted from one
member of the aforementioned family to another. We perform a phase space
reduction using inflying coordinates with an unspecified boosting function,
resulting in a reduced theory with residual coordinate freedom. Finally, we
discuss quantization, and comment on the utility of the reduced system for the
study of coordinate effects and the role of observers in quantum gravity.
