  We say that two permutations $\pi$ and $\rho$ have separated descents at
position $k$ if $\pi$ has no descents before position $k$ and $\rho$ has no
descents after position $k$. We give a counting formula, in terms of reduced
word tableaux, for computing the structure constants of products of Schubert
polynomials indexed by permutations with separated descents, and recognize that
these structure constants are certain Edelman-Greene coefficients. Our approach
uses generalizations of Sch\"utzenberger's jeu de taquin algorithm and the
Edelman-Greene correspondence via bumpless pipe dreams.
