  We say two posets are "doppelg\"angers" if they have the same number of
$P$-partitions of each height $k$. We give a uniform framework for bijective
proofs that posets are doppelg\"angers by synthesizing $K$-theoretic Schubert
calculus techniques of H. Thomas and A. Yong with M. Haiman's rectification
bijection and an observation of R. Proctor. Geometrically, these bijections
reflect the rational equivalence of certain subvarieties of minuscule flag
manifolds. As a special case, we provide the first bijective proof of a 1983
theorem of R. Proctor---that plane partitions of height $k$ in a rectangle are
equinumerous with plane partitions of height $k$ in a trapezoid.
