  We study topological phases in one-dimensional open Floquet systems driven by
chiral symmetric nonunitary time evolution. We derive a procedure to calculate
topological numbers from nonunitary time-evolution operators with chiral
symmetry. While the procedure has been applied to open Floquet systems
described by nonunitary time-evolution operators, we give the microscopic
foundation and clarify its validity for the first time. We construct a model of
chiral symmetric nonunitary quantum walks classified into class BDI$^\dagger$
or AIII, which is one of enlarged symmetry classes for topological phases in
open systems, based on experiments of discrete-time quantum walks. Then, we
confirm that the topological numbers obtained from the derived procedure give
correct predictions of the emergent edge states. We also show that the model
retains $\mathcal{PT}$ symmetry in certain cases and its dynamics is crucially
affected by the presence or absence of $\mathcal{PT}$ symmetry.
