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The Paris Agreement legally commits the international community to keep anthropogenic global warming well below 2.0 °C, while efforts shall be made to hold the 1.5 °C-line. Under climate business as usual, however, the transgression of those lines will happen in the next few decades, causing major adaptation challenges. Fully-fledged Earth System Models are usually employed for concrete overshoot-timing projections, yet they are not only computationally expensive, but their internal variability is model dependent, which may significantly distort (and invalidate) their projections. Here we present, as an alternative, a purely data-driven approach based on the persistence properties of the observed global temperatures. We quantify, in a probabilistic way, the natural variability that must be superimposed on the anthropogenic trends in order to retrieve the observed warming behavior. When assuming that the anthropogenic warming continues at the current rate, we actually arrive at comparable overshoot timing estimates as the Earth System Models and provide an explanation for this finding. Since the two approaches are independent, they support each other strongly and highlight the need for an effective overshoot management.