  We propose a one-dimensional Floquet ladder that possesses two distinct
topological transport channels with opposite directionality. The transport
channels occur due to a $\mathbb Z_2$ non-Hermitian Floquet topological phase
that is protected by time-reversal symmetry. The signatures of this phase are
two pairs of Kramers degenerate Floquet quasienergy bands that are separated by
an imaginary gap. We discuss how the Floquet ladder can be implemented in a
photonic waveguide lattice and show that the direction of transport in the
resulting waveguide structure can be externally controlled by focusing two
light beams into adjacent waveguides. The relative phase between the two light
beams selects which of the two transport channels is predominantly populated,
while the angles of incidence of the two light beams determine which of the
transport channels is suppressed by non-Hermitian losses. We identify the
optimal lattice parameters for the external control of transport and
demonstrate the robustness of this mechanism against disorder.
