We study phases and structures of mixtures of two representative chromonic liquid crystal materials, sunset yellow FCF (SSY) and disodium cromoglycate (DSCG), in water. A variety of combinations of isotropic, nematic (N), and columnar (also called M) phases are observed depending on their concentrations, and a phase diagram is made. We find a tendency for DSCG-rich regions to show higher-order phases while SSY-rich regions show lower-order ones. We observe uniform mesophases only when one of the materials is sparse in the N phases. Their miscibility in M phases is so low that essentially complete phase separation occurs. X-ray scattering and spectroscopy studies confirm that SSY and DSCG molecules do not mix when they form chromonic aggregates and neither do their aggregates when they form M phases.