In many-body quantum physics, the global state encodes all correlations, but the information most naturally available is often local. This makes it important to understand when a small collection of reduced density matrices already fixes the whole state. Such uniqueness questions are especially important for tomography and for the study of entanglement, because agreement with local data from a pure state does not by itself exclude a different mixed global state. We prove a generic uniqueness theorem for finite-dimensional tripartite systems. Let HA, HB, HC be complex Hilbert spaces with dim HA >= 2, and consider pure states on HA HB HC. Then almost every such pure state is uniquely determined among all states by the two overlapping two-party marginals AB and AC. The two overlapping marginals therefore form a complete local fingerprint of a generic pure state: they determine the unmeasured (BC) marginal and the expectation value of every global observable, while excluding all mixed global explanations compatible with the same local data. Previous results were restricted either to uniqueness against pure competitors or to uniqueness against all competitors under stronger dimension assumptions or balanced subsystem dimensions. Our result removes both restrictions in the strongest possible way.