Recent observations of the Lyman-alpha forest show large-scale spatial variations in the intergalactic Lyman-alpha opacity that grow rapidly with redshift at z>5, far in excess of expectations from empirically motivated models. Previous studies have attempted to explain this excess with spatial fluctuations in the ionizing background, but found that this required either extremely rare sources or problematically low values for the mean free path of ionizing photons. Here we report that much -- or potentially all -- of the observed excess likely arises from residual spatial variations in temperature that are an inevitable byproduct of a patchy and extended reionization process. The amplitude of opacity fluctuations generated in this way depends on the timing and duration of reionization. If the entire excess is due to temperature variations alone, the observed fluctuation amplitude favors a late-ending but extended reionization process that was roughly half complete by z~9 and that ended at z~6. In this scenario, the highest opacities occur in regions that reionized earliest, since they have had the most time to cool, while the lowest opacities occur in the warmer regions that reionized most recently. This correspondence potentially opens a new observational window into patchy reionization.
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