single-cell sequencing methods have enabled the profiling of multiple types of molecular readouts at cellular resolution, and recent developments in spatial barcoding, in situ hybridization, and in situ sequencing allow such molecular readouts to retain their spatial context. Since no technology can provide complete characterization across all layers of biological modalities within the same cell, there is pervasive need for computational cross-modal integration (also called diagonal integration) of single-cell and spatial omics data. For current methods, the feasibility of cross-modal integration relies on the existence of highly correlated, a priori "linked" features. When such linked features are few or uninformative, a scenario that we call "weak linkage", existing methods fail. We developed MaxFuse, a cross-modal data integration method that, through iterative co-embedding, data smoothing, and cell matching, leverages all information in each modality to obtain high-quality integration. MaxFuse is modality-agnostic and, through comprehensive benchmarks on single-cell and spatial ground-truth multiome datasets, demonstrates high robustness and accuracy in the weak linkage scenario. A prototypical example of weak linkage is the integration of spatial proteomic data with single-cell sequencing data. On two example analyses of this type, we demonstrate how MaxFuse enables the spatial consolidation of proteomic, transcriptomic and epigenomic information at single-cell resolution on the same tissue section.