  Periodically driven non-Hermitian systems could possess exotic nonequilibrium
phases with unique topological, dynamical and transport properties. In this
work, we introduce an experimentally realizable two-leg ladder model subjecting
to both time-periodic quenches and non-Hermitian effects, which belongs to an
extended CII symmetry class. Due to the interplay between drivings and
nonreciprocity, rich non-Hermitian Floquet topological phases emerge in the
system, with each of them been characterized by a pair of even-integer
topological invariants $(w_{0},w_{\pi})\in2\mathbb{Z}\times2\mathbb{Z}$. Under
the open boundary condition, these invariants further predict the number of
zero- and $\pi$-quasienergy modes localized around the edges of the system. We
finally construct a generalized version of the mean chiral displacement, which
could be employed as a dynamical probe to the topological invariants of
non-Hermitian Floquet phases in the CII symmetry class. Our work thus
introduces a new type of non-Hermitian Floquet topological matter, and further
reveals the richness of topology and dynamics in driven open systems.
