Feynman diagrams (notably the triangle diagram) involving heavy enough particles contain branch cuts on the physical sheet-anomalous thresholds-which, unlike normal thresholds and bound-state poles, do not correspond to any asymptotic n-particle state. "Who ordered that?" We show that anomalous thresholds arise as a consequence of established S-matrix principles and two reasonable assumptions: unitarity below the physical region and analyticity in the mass. We find explicit nonperturbative formulas for the anomalous threshold singularity and test them against the Coleman-Thun poles of the E8 integrable model.