  New families of non-parity-time-symmetric complex potentials with all-real
spectra are derived by the supersymmetry method and the pseudo-Hermiticity
method. With the supersymmetry method, we find families of
non-parity-time-symmetric complex partner potentials which share the same
spectrum as base potentials with known real spectra, such as the (complex)
Wadati potentials. Different from previous supersymmetry derivations of
potentials with real spectra, our derivation does not utilize discrete
eigenmodes of base potentials. As a result, our partner potentials feature
explicit analytical expressions which contain free functions. With the
pseudo-Hermiticity method, we derive a new class of non-parity-time-symmetric
complex potentials with free functions and constants, whose eigenvalues appear
as conjugate pairs. This eigenvalue symmetry forces the spectrum to be all-real
for a wide range of choices of these functions and constants in the potential.
Tuning these free functions and constants, phase transition can also be
induced, where conjugate pairs of complex eigenvalues emerge in the spectrum.
