In image compression, with recent advances in generative modeling, existence of a trade-off between the rate and perceptual quality has been brought to light, where the perceptual quality is measured by the closeness of the output and source distributions. We consider the compression of a memoryless source sequence $X^n=(X_1, \ldots, X_n)$ in the presence of memoryless side information $Z^n=(Z_1, \ldots, Z_n),$ originally studied by Wyner and Ziv, but elucidate the impact of a strong perfect realism constraint, which requires the joint distribution of output symbols $Y^n=(Y_1,...,Y_n)$ to match the distribution of the source sequence. We consider two cases: when $Z^n$ is available only at the decoder, or at both the encoder and decoder, and characterize the information theoretic limits under various scenarios. Previous works show the superiority of randomized codes under strong perceptual quality constraints. When $Z^n$ is available at both terminals, we characterize its dual role, as a source of common randomness, and as a second look on the source for the receiver. We also study different notions of strong perfect realism which we call marginal realism, joint realism and near-perfect realism. We derive explicit solutions when $X$ and $Z$ are jointly Gaussian under the squared error distortion measure. In traditional lossy compression, having $Z$ only at the decoder imposes no rate penalty in the Gaussian scenario. We show that, when strong perfect realism constraints are imposed this holds only when sufficient common randomness is available.