  We demonstrate that the existence of a Hermitian time-dependent intertwining
operator that maps the non-Hermitian time-dependent energy operator to its
Hermitian conjugate and its right to its left eigenstates guarantees the
reality of the instantaneous energies. This property holds throughout all three
$\cal{PT}$-regimes, in the time-independent scenario referred to as the
$\cal{PT}$-symmetric regime, the exceptional point and 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, instead of the usually used eigenfunctions
of the Hamiltonian. We show that this proposal always leads to real Berry
phases. We illustrate the working of our general proposals with two explicit
examples for a time-dependent non-Hermitian spin model.
