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We present an encoding of a polynomial system into vanishing andnon-vanishing constraints on almost-principal minors of a symmetric,principally regular matrix, such that the solvability of the system over somefield is equivalent to the satisfiability of the constraints over that field.This implies two complexity results about Gaussian conditional independencestructures. First, all real algebraic numbers are necessary to constructinhabitants of non-empty Gaussian statistical models defined by conditionalindependence and dependence constraints. This gives a negative answer to aquestion of Petr \v{S}ime\v{c}ek. Second, we prove that the implication problemfor Gaussian CI is polynomial-time equivalent to the existential theory of thereals.

Incidence geometry in the projective plane via almost-principal minors of symmetric matrices