  In this article we study 3D non-Hermitian higher-order Dirac semimetals
(NHHODSMs). Our focus is on $C_4$-symmetric non-Hermitian systems where we
investigate inversion ($\mathcal{I}$) or time-reversal ($\mathcal{T}$)
symmetric models of NHHODSMs having real bulk spectra. We show that they
exhibit the striking property that the bulk and surfaces are anti-PT and PT
symmetric, respectively, and so belong to two different topological classes
realizing a novel non-Hermitian topological phase which we call a
\emph{hybrid-PT topological phases}. Interestingly, while the bulk spectrum is
still fully real, we find that exceptional Fermi-rings (EFRs) appear connecting
the two Dirac nodes on the surface. This provides a route to probe and utilize
both the bulk Dirac physics and exceptional rings/points on equal footing.
Moreover, particularly for $\mathcal{T}$-NHHODSMs, we also find real hinge-arcs
connecting the surface EFRs. We show that this higher-order topology can be
characterized using a biorthogonal real-space formula of the quadrupole moment.
Furthermore, by applying Hermitian $C_4$-symmetric perturbations, we discover
various novel phases, particularly: (i) an intrinsic $\mathcal{I}$-NHHODSM
having hinge arcs and gapped surfaces, and (ii) a novel $\mathcal{T}$-symmetric
skin-topological HODSM which possesses both topological and skin hinge modes.
The interplay between non-Hermition and higher-order topology in this work
paves the way toward uncovering rich phenomena and hybrid functionality that
can be readily realized in experiment.
