  The Shapiro conjecture in the real Schubert calculus fails to hold for flag
manifolds, but in a very interesting way. In this extended abstract, we give a
refinement of that conjecture for the flag manifold and present massive
experimentation (over 12 GigaHertz-Years) that supports our refined conjecture.
We also establish relationships between different cases of the conjecture and
describe some new phenomena uncovered in this experimentation.
