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

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