  Parity-time ($\mathcal{PT}$) symmetry plays an important role both in
non-Hermitian and topological systems. In non-Hermitian systems $\mathcal{PT}$
symmetry can lead to an entirely real energy spectrum, while in topological
systems $\mathcal{PT}$ symmetry gives rise to stable and protected Dirac
points. Here, we study a $\mathcal{PT}$-symmetric system which is both
non-Hermitian and topological, namely a $\mathcal{PT}$-symmetric Dirac
semimetal with non-Hermitian perturbations in three dimensions. We find that,
in general, there are only two types of symmetry allowed non-Hermitian
perturbations, namely non-Hermitian kinetic potentials, and non-Hermitian
anti-commuting potentials. For both of these non-Hermitian potentials we
investigate the band topology for open and periodic boundary conditions,
determine the exceptional points, and compute the surface states. We find that
with periodic boundary conditions, the non-Hermitian kinetic potential leads to
exceptional rings, while the non-Hermitian anti-commuting potential generates
exceptional spheres, which separate regions with broken $\mathcal{PT}$ symmetry
from regions with unbroken $\mathcal{PT}$ symmetry. With open boundary
conditions, the non-Hermitian kinetic potential induces a non-Hermitian skin
effect which is localized on both sides of the sample due to symmetry, while
the non-Hermitian anticommuting potential leads to Fermi ribbon surface states.
