We present a general method for constructing real solutions to some problems
in enumerative geometry which gives lower bounds on the
maximum number of real solutions.
We apply this method to show that two new classes of enumerative
geometric problems on flag manifolds may have all their solutions be real
and modify this method to show that another class may have no real
solutions, which is a new phenomenon.
This method originated in a numerical homotopy
continuation algorithm adapted to the special Schubert calculus on
Grassmannians
and in principle gives optimal numerical homotopy algorithms for finding
explicit solutions to these other enumerative problems.