Coupled XY model systems consisting of three-dimensional (3D) systems with disordered interlayer physics are of significant theoretical interest. We realize a set of coupled quasi-2D layers of Rb-87 in the presence of disordered interlayer coupling. This is achieved with our high bandwidth arbitrary optical lattice to obviate restrictions on the dimensionality of disorder with speckle-generated optical fields. We identify phase crossover regions compatible with the existence of a pair of intermediate Griffiths phases between a thermal state and the emergence of bulk 3D superfluidity.