In this Letter, we study three-dimensional 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-$\mathcal{P}\mathcal{T}$ and $\mathcal{P}\mathcal{T}$ symmetric, respectively, and so belong to two different topological classes realizing a non-Hermitian topological phase which we call a hybrid-$\mathcal{P}\mathcal{T}$ topological phase. 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 phases, particularly (i) an intrinsic $\mathcal{I}$-NHHODSM having hinge arcs and surface exceptional Fermi arcs, and (ii) a $\mathcal{T}$-symmetric skin-topological HODSM which possesses both topological and skin hinge modes. The interplay between non-Hermition and higher-order topology in this Letter paves the way toward uncovering rich phenomena and hybrid functionality that can be readily realized in experiment.

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• Exotic phases of matter
• Topological phases of matter
• Dirac semimetal
• Semimetals
• Topological materials