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Here we address a long-standing puzzle of ice-age climate called the "fly in the ointment of the Milankovitch theory." Using geomorphic mapping and 10 Be surface-exposure dating, we show that five moraine belts were formed during maxima of the last ice age by the Pukaki glacier in New Zealand's Southern Alps. They afford ages of 41.76 ± 1.09 ka, 35.50 ± 1.26 ka, 27.17 ± 0.68 ka, 20.27 ± 0.60 ka, and 18.29 ± 0.49 ka. These five maxima spanned an entire precessional cycle in summer insolation intensity at the latitude of the Southern Alps. A similar mismatch between summer insolation and glacier extent also characterized the Chilean Lake District in the mid-latitudes of South America. Thus, in apparent contrast to northern ice sheets linked by Milankovitch to summer insolation at 65°N latitude, the behavior of southern mid-latitude glaciers was not tied to local summer insolation intensity. Instead, glacier extent between 41.76 ka and 18.29 ka, as well as during the last termination, was aligned with Southern Ocean surface temperature and with atmospheric carbon dioxide.