Integrating datasets from different disciplines is hard because the data are often qualitatively different in meaning, scale, and reliability. When two datasets describe the same entities, many scientific questions can be phrased around whether the (dis)similarities between entities are conserved across such different data. Our method, CLARITY, quantifies consistency across datasets, identifies where inconsistencies arise, and aids in their interpretation. We illustrate this using three diverse comparisons: gene methylation vs expression, evolution of language sounds vs word use, and country-level economic metrics vs cultural beliefs. The non-parametric approach is robust to noise and differences in scaling, and makes only weak assumptions about how the data were generated. It operates by decomposing similarities into two components: a `structural' component analogous to a clustering, and an underlying `relationship' between those structures. This allows a `structural comparison' between two similarity matrices using their predictability from `structure'. Significance is assessed with the help of re-sampling appropriate for each dataset. The software, CLARITY, is available as an R package from https://github.com/danjlawson/CLARITY.