  The Shapiro conjecture in the real Schubert calculus, while likely true for
Grassmannians, fails to hold for flag manifolds, but in a very interesting way.
We give a refinement of the Shapiro conjecture for the flag manifold and
present massive (15.76 gigahertz-years) computational experimentation in
support of this refined conjecture.
  We also prove the conjecture in some special cases using discriminants and
establish relationships between different cases of the conjecture.
