  We study non-Hermitian Josephson junctions formed by superconductors with a
finite phase difference under non-Hermiticity, which naturally appears due to
coupling to normal reservoirs. Depending on the structure of non-Hermiticity,
captured in terms of retarded self-energies, the low-energy spectrum hosts
topologically stable exceptional points either at zero or finite real energies
as a function of the superconducting phase difference. Interestingly, the
corresponding phase-biased supercurrents may acquire divergent profiles at such
exceptional points. This instance is a natural and unique non-Hermitian effect
that signals a possible way to enhance the sensitivity of Josephson junctions.
Our work opens a way for realizing unique non-Hermitian phenomena due to the
interplay between non-Hermitian topology and the Josephson effect.
