  While parity-time (PT)-symmetric systems can exhibit real spectra in the
exact PT-symmetry regime, the PT-symmetry is actually not a necessary condition
for the real spectra. Here we show that non-PT-symmetric photonic crystals
carrying Dirac-like cone dispersions can always exhibit real spectra as long as
the average non-Hermiticity strength within the unit cell for the eigenstates
is zero. By building a non-Hermitian Hamiltonian model, we find that the real
spectra of the non-PT-symmetric system can be explained using the concept of
pseudo-Hermiticity. We demonstrate using effective medium theories that in the
long-wavelength limit, such non-PT-symmetric photonic crystals behave like the
so-called complex conjugate medium (CCM) whose refractive index is real but
whose permittivity and permeability are complex numbers. The real refractive
index for this effective CCM is guaranteed by the real spectrum of the photonic
crystals and the complex permittivity and permeability comes from the
non-PT-symmetric loss-gain distributions. We show some interesting phenomena
associated with the CCM, such as the lasing effect.
