We study a central problem in multimodal neuroimaging analysis, i.e., identification of significantly correlated brain regions between multiple imaging modalities. We propose a spatially-varying correlation model and the associated inference procedure, which improves substantially over the common alternative solutions of voxel-wise and region-wise analysis. Compared to voxel-wise analysis, our method aggregates voxels with similar correlations into regions, takes into account spatial continuity of correlations at nearby voxels, and enjoys a much higher detection power. Compared to region-wise analysis, our method does not rely on any pre-specified brain region map, but instead finds homogenous correlation regions adaptively given the data. We applied our method to a multimodal positron emission tomography study, and found brain regions with significant correlation between tau and glucose metabolism that voxel-wise or region-wise analysis failed to identify. Our findings conform and lend additional support to prior hypotheses about how pathological proteins of Alzheimer's disease, tau and amyloid, interact with glucose metabolism in the aging human brain.