We describe a partition of the double flag variety G/B+ × G/B- of a complex semisimple algebraic group G analogous to the Deodhar partition on the flag variety G/B+. This partition is a refinement of the stratification into orbits both for B+ × B- and for the diagonal action of G, just as Deodhar's partition 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+ × G/B-, which would generalize results of Fomin and Zelevinsky on double Bruhat cells and Marsh and Rietsch on double Schubert cells.