  We provide a possible fully geometric formulation of the core idea of quantum
reference frames (QRFs) as it has been applied in the context of gravity,
freeing its definition from unnecessary (though convenient) ingredients, such
as coordinate systems. Our formulation is based on two main ideas. First, a QRF
encodes uncertainty about what is the observer's (and, hence, the measuring
apparatus's) perception of time and space at each spacetime point (i.e.,
event). For this, an observer at an event $p$ is modeled, as usual, as a tetrad
in the tangent space $T_p$. So a QRF at an event $p$ is a complex function on
the tetrads at $p$. Second, we use the result that one can specify a metric on
a given manifold by stipulating that a basis one assigns at each tangent space
is to be a tetrad in the metric one wants to specify. Hence a spacetime, i.e.
manifold plus metric, together with a choice of "point of view" on it, is
represented by a section of the bundle of bases, understood as taking the basis
assigned to each point to be a tetrad. Thus a superposition of spacetimes gets
represented as, roughly speaking, an assignment of complex amplitudes to
sections of this bundle. A QRF, defined here as the collection of complex
amplitudes assigned to bases at events--i.e., a complex function defined on the
bundle of bases of the manifold--can describe, in a local way (i.e.,
attributing the amplitudes to bases at events instead of to whole sections),
these superpositions.
  We believe that this formulation sheds some light on some conceptual aspects
and possible extensions of current ideas about QRFs. For instance, thinking in
geometric terms makes it clear that the idea of QRFs applied to the
gravitational scenarios treated in the literature (beyond linear approximation)
lacks predictive power due to arbitrariness which, we argue, can only be
resolved by some further input from physics.
