  The slack ideal of a polytope is a saturated determinantal ideal that gives
rise to a new model for the realization space of the polytope. The simplest
slack ideals are toric and have connections to projectively unique polytopes.
We prove that if a projectively unique polytope has a toric slack ideal, then
it is the toric ideal of the bipartite graph of vertex-facet non-incidences of
the polytope. The slack ideal of a polytope is contained in this toric ideal if
and only if the polytope is morally 2-level, a generalization of the 2-level
property in polytopes. We show that polytopes that do not admit rational
realizations cannot have toric slack ideals. A classical example of a
projectively unique polytope with no rational realizations is due to Perles. We
prove that the slack ideal of the Perles polytope is reducible, providing the
first example of a slack ideal that is not prime.
