  An AdS eternal black hole in equilibrium with a finite temperature bath
presents a Hawking-like information paradox due to a continuous exchange of
radiation with the bath. The non-perturbative gravitational effect, the replica
wormhole, cures this paradox by introducing a non-trivial entanglement wedge
for the bath after Page time. In this paper, we analyse the theory dependence
of this non-perturbative effect by randomising the boundary conditions of some
of the bulk matter fields. We explicitly analyse this in JT gravity by
introducing a matter CFT in the AdS region with random boundary conditions at
the AdS boundary that are drawn from a distribution. Using the island formula
and the extended strong subadditivity due to Carlen and Lieb, we show that at
late times the black hole interior is contained inside the entanglement wedge
of a reference Hilbert space that encodes the information about the random
boundary conditions. Consequently, the reconstruction of the black hole
interior from the radiation, in particular the region near the singularity,
requires a detailed knowledge of the theory.
