  We show that the totally nonnegative part of the twisted product of flag
varieties of a Kac-Moody group admits a cellular decomposition, and the closure
of each cell is a topological manifold with boundary. We also establish
explicit parameterizations of each totally positive cell.
  In the special cases of double flag varieties and braid varieties, we show
that the totally nonnegative parts are regular CW complexes homeomorphic to
closed balls. Moreover, we prove that the link of any totally nonnegative
double Bruhat cell in a reductive group is a regular CW complex homeomorphic to
a closed ball, solving an open problem of Fomin and Zelevinsky.
