  We investigate the transport properties of a bilayer exciton condensate that
is contacted by four superconducting leads. We focus on the equilibrium regime
and investigate how the Josephson currents induced in the bilayer by phase
biases applied to the superconducting electrodes are affected by the presence
of an exciton condensate in the bulk of the system. As long as the distance
between the superconducting electrodes is much larger than the exciton
coherence length, the Josephson current depends only on the difference between
the phase biases in the two layers. This result holds true in both short- and
long-junction limits. We relate it to a novel correlated four-particle Andreev
process which occurs at the superconductor - exciton condensate interface. The
system we investigate provides an implementation of the supercurrent mirror
proposed by Kitaev as a viable way to realize topologically protected qubits.
