The generation mechanism of screech harmonics in supersonic exhausts is revealed using shadowgraph imaging and acoustic far-field measurements for a rectangular, high aspect-ratio nozzle. The coherent information associated with screech and its harmonics, i.e. flow structures and acoustic radiation pattern, is extracted from the time-resolved shadowgraph images. We show that, for large lateral distortions of the jet plume, the passage of screech associated flow structures triggers the screech-cyclic formation of shocks, which travel downstream and merge with the original shocks. The interaction of the shock waves with the flow structures associated with screech alters the appearance of the perturbations in the mixing layer, which constitute the higher harmonics of screech. Visualisations of the acoustic radiation pattern expose that the third and higher screech tone harmonics originate from these interaction locations. Further, the occurrence of mode resonance between the screech and its harmonics is demonstrated, where the mode resonance location coincides with the screech tone origin.