  We introduce two new bases for polynomials that lift monomial and fundamental
quasisymmetric functions to the full polynomial ring. By defining a new
condition on pipe dreams, called quasi-Yamanouchi, we give a positive
combinatorial rule for expanding Schubert polynomials into these new bases that
parallels the expansion of Schur functions into fundamental quasisymmetric
functions. As a result, we obtain a refinement of the stable limits of Schubert
polynomials to Stanley symmetric functions. We also give combinatorial rules
for the positive structure constants of these bases that generalize the
quasi-shuffle product and shuffle product, respectively. We use this to give a
Littlewood--Richardson rule for expanding a product of Schubert polynomials
into fundamental slide polynomials and to give formulas for products of Stanley
symmetric functions in terms of Schubert structure constants.
