  In their study of infinite flag varieties, Lam, Lee, and Shimozono (2021)
introduced bumpless pipe dreams in a new combinatorial formula for double
Schubert polynomials. These polynomials are the TxT-equivariant cohomology
classes of matrix Schubert varieties and of their flat degenerations. We give
diagonal term orders with respect to which bumpless pipe dreams index the
irreducible components of diagonal Gr\"obner degenerations of matrix Schubert
varieties, counted with scheme-theoretic multiplicity.
  This indexing was conjectured by Hamaker, Pechenik, and Weigandt (2022). We
also give a generalization to equidimensional unions of matrix Schubert
varieties. This result establishes that bumpless pipe dreams are dual to and as
geometrically natural as classical pipe dreams, for which an analogous
anti-diagonal theory was developed by Knutson and Miller (2005).
