We demonstrate that the existence of a time-dependent antilinear symmetry
operator of the non-Hermitian energy operator that maps its right to its
conjugate left eigenstates guarantees the reality of the instantaneous
energies. This property holds throughout all three $\cal{PT}$-regimes, these
are what in the time-independent scenario are referred to as the
$\cal{PT}$-symmetric regime, the exceptional point as well as the spontaneously
broken $\cal{PT}$-regime. We also propose a modified adiabatic approximation
consisting of an expansion of the wavefunctions in terms the instantaneous
eigenstates of the energy operator, rather than of those of the Hamiltonian,
that will always lead to real Berry phases. We illustrate the working of our
general proposals with two explicit examples for a time-dependent non-Hermitian
spin model.
