  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.
