  Non-Hermitian Hamiltonians provide a simple picture for analyzing systems
with natural or induced gain and loss; however, in general, such Hamiltonians
feature complex energies and a corresponding non-orthonormal eigenbasis.
Provided that the Hamiltonian has $\mathcal{PT}$ symmetry, it is possible to
find a regime in which the eigenspectrum is completely real. In the case of
static $\mathcal{PT}$-symmetric extensions of the simple Su-Schrieffer-Heeger
model, it has been shown that the energies associated with any edge states are
guaranteed to be complex. Moving to a time-dependent system means that
treatment of the Hamiltonian must be done at the effective time-scale of the
modulation itself, allowing for more intricate phases to occur than in the
static case. It has been demonstrated that with particular classes of periodic
driving, achieving a real topological phase at high driving frequency is
possible. In the present paper, we show the details of this process by using a
simple two-step periodic modulation. We obtain a rigorous expression for the
effective Floquet Hamiltonian and compare its symmetries to those of the
original Hamiltonians which comprise the modulation steps. The $\mathcal{PT}$
phase of the effective Hamiltonian is dependent on the modulation frequency as
well as the gain/loss strength. Furthermore, the topologically nontrivial
regime of the $\mathcal{PT}$-unbroken phase admits highly-localized edge states
with real eigenvalues in both the high frequency case and below it, albeit
within a smaller extent of the parameter space.
