  In holographic theories, the reflected entropy has been shown to be dual to
the area of the entanglement wedge cross section. We study the same problem in
random tensor networks demonstrating an equivalent duality. For a single random
tensor we analyze the important non-perturbative effects that smooth out the
discontinuity in the reflected entropy across the Page phase transition. By
summing over all such effects, we obtain the reflected entanglement spectrum
analytically, which agrees well with numerical studies. This motivates a
prescription for the analytic continuation required in computing the reflected
entropy and its R\'enyi generalization which resolves an order of limits issue
previously identified in the literature. We apply this prescription to
hyperbolic tensor networks and find answers consistent with holographic
expectations. In particular, the random tensor network has the same non-trivial
tripartite entanglement structure expected from holographic states. We
furthermore show that the reflected R\'enyi spectrum is not flat, in sharp
contrast to the usual R\'enyi spectrum of these networks. We argue that the
various distinct contributions to the reflected entanglement spectrum can be
organized into approximate superselection sectors. We interpret this as
resulting from an effective description of the canonically purified state as a
superposition of distinct tensor network states. Each network is constructed by
doubling and gluing various candidate entanglement wedges of the original
network. The superselection sectors are labelled by the different
cross-sectional areas of these candidate entanglement wedges.
