  Multitriangulations, and more generally subword complexes, yield a large
family of simplicial complexes that are homeomorphic to spheres. Until now, all
attempts to prove or disprove that they can be realized as convex polytopes
faced major obstacles. In this article, we lay out the foundations of a
framework -- built upon notions from algebraic combinatorics and discrete
geometry -- that allows a deeper understanding of geometric realizations of
subword complexes of Coxeter groups. Namely, we describe explicitly a family of
chirotopes that encapsulate the necessary information to obtain geometric
realizations of subword complexes. Further, we show that the space of geometric
realizations of this family covers that of subword complexes, making this
combinatorially defined family into a natural object to study.
  The family of chirotopes is described through certain parameter matrices.
That is, given a finite Coxeter group, we present matrices where certain minors
have prescribed signs. Parameter matrices are universal: The existence of these
matrices combined with conditions in terms of Schur functions is equivalent to
the realizability of all subword complexes of this Coxeter group as chirotopes.
Finally, parameter matrices provide extensions of combinatorial identities; for
instance, the Vandermonde determinant and the dual Cauchy identity are
recovered through suitable choices of parameters.
