  Brown and Gordon asked whether the Poisson Dixmier-Moeglin equivalence holds
for any complex affine Poisson algebra; that is, whether the sets of Poisson
rational ideals, Poisson primitive ideals, and Poisson locally closed ideals
coincide. In this article a complete answer is given to this question using
techniques from differential-algebraic geometry and model theory. In
particular, it is shown that while the sets of Poisson rational and Poisson
primitive ideals do coincide, in every Krull dimension at least four there are
complex affine Poisson algebras with Poisson rational ideals that are not
Poisson locally closed. These counterexamples also give rise to counterexamples
to the classical (noncommutative) Dixmier-Moeglin equivalence in finite
$\operatorname{GK}$ dimension. A weaker version of the Poisson Dixmier-Moeglin
equivalence is proven for all complex affine Poisson algebras, from which it
follows that the full equivalence holds in Krull dimension three or less.
Finally, it is shown that everything, except possibly that rationality implies
primitivity, can be done over an arbitrary base field of characteristic zero.
