I will send you the written version later and I will state it now. Take an open Schubert cell in the space of complete flags which corresponds to uppertriangular matrices with 1's on the diagonal. Take any flag in this cell (say, corresponding to the identity matrix). Consider its train, i.e. the union of all Schubert cells of positive codimension. The complement to its train contains several connected components, in particular, one of the connected components is a semigroup coinciding with the set of all totally positive uppertriangular matrices (i.e. all minors which can be positive are positive). If you choose another initial complete flag you still have a semigroup of 'totally positive matrices associated to this flag which is obtained by multiplication.

Conjecture. If you take a sequence f(1),...,f(k) of complete flags in the open Schubert cell such that f(i) belongs to the 'totally positive' semigroup of f(i-1) (and therefore of all the previous ones) and state the necessary number of Schubert conditions at these flags you will get a totally real problem in Schubert calculus.

Yours B.Sh.    Unfortunately, this conjecture has been found to be false, and so this is no longer relevant.