Mazur has drawn attention to the question of determining when the cohomology of a smooth, projective variety over a number field can be modeled by an abelian variety. I will discuss recent work with Casalaina-Martin and Vial which constructs such a "phantom" abelian variety for varieties with maximal geometric coniveau. In the special case of cohomology in degree three, we show that the image of the (complex) Abel-Jacobi map admits a distinguished model over the base field, and that an algebraic correspondence realizes this descended intermediate Jacobian as a phantom.