Significance Ice streams form the backbone of the flow field of ice sheets. They are known to exhibit a complex spatiotemporal dynamics, which is largely not well understood. Understanding the controls on such dynamics is crucial to sea level change projections as well as to the interpretation of paleorecords. Our contribution unravels the impact of climate variability on the temporal dynamics of ice streams. In particular, we show that minimal, and at the same time realistic, random fluctuations due to climate variability are capable of producing unprecedented behaviors in ice stream dynamics. These results may open a new perspective on the past and future behavior of ice sheets.