  We describe a stratification on the double flag variety $G/B^+\times G/B^-$
of a complex semisimple algebraic group $G$ analogous to the Deodhar
stratification on the flag variety $G/B^+$, which is a refinement of the
stratification into orbits both for $B^+\times B^-$ and for the diagonal action
of $G$, just as Deodhar's stratification refines the orbits of $B^+$ and $B^-$.
  We give a coordinate system on each stratum, and show that all strata are
coisotropic subvarieties. Also, we discuss possible connections to the positive
and cluster geometry of $G/B^+\times G/B^-$, which would generalize results of
Fomin and Zelevinsky on double Bruhat cells and Marsh and Rietsch on double
Schubert cells.
