Since a tetrad encodes a timelike direction, three orthogonal spatial directions and units, it gives a "configuration" in spacetime of an idealized infinitesimal apparatus ("rulers and a clock"). So a natural "first guess" is that the state of a quantum reference frame should be an assignment of complex amplitudes to a set of tetrads. We develop this idea in two ways: both of them in terms of bundles over spacetime of bases of the tangent spaces. 

The first way considers a single spacetime geometry, so that a quantum reference frame gives a superposition of perspectives on that geometry. The second way considers different spacetime geometries, which implies that a quantum reference frame can describe a superposition of perspectives on a superposition of geometries—not "just" a superposition of perspectives on a single geometry. 

The main mathematical contrast between these two ways is that: (i) the first way uses the orthonormal frame bundle, for the given spacetime geometry; while (ii) the second way uses the frame bundle in the usual sense of containing all bases of all spacetime points' tangent spaces.