  We introduce a new method for finding a non-realizability certificate of a
simplicial sphere Sigma: we exhibit a monomial combination of classical 3-term
Pl\"ucker relations that yields a sum of products of determinants that are
known to be positive in any realization of Sigma; but their sum should vanish,
contradiction. Using this technique, we prove for the first time the
non-realizability of a balanced 2-neighborly 3-sphere constructed by Zheng, a
family of highly neighborly centrally symmetric spheres constructed by by Novik
and Zheng, and several combinatorial prismatoids introduced by Criado and
Santos. The method in fact works for orientable pseudo-manifolds, not just for
spheres.
