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The characterisation of the multiplicity of high-mass stars is of fundamental importance to understand their evolution, the diversity of observed core-collapse supernovae and the formation of gravitational wave progenitor systems. Despite that, until recently, one of the final phases of massive star evolution -- the cool supergiant phase -- has received comparatively little attention. In this study we aim to explore the multiplicity among the cool supergiants (CSGs) in the Large and Small Magellanic Clouds (LMC and SMC, respectively). To do this we compile extensive archival radial velocity (RV) measurements for over 1000 CSGs from the LMC and SMC, spanning a baseline of over 40 years. By statistically correcting the RV measurements of each stellar catalogue to the Gaia DR2 reference frame we are able to effectively compare these diverse observations. We identify 45 CSGs where RV variations cannot be explained through intrinsic variability, and are hence considered binary systems. We obtain a minimum binary fraction of $15\pm4\%$ for the SMC and of $14\pm5\%$ for the LMC. Combining these results, we determine a minimum binary fraction of $15\pm3\%$ for CSGs. These results are in good agreement with previous results which apply a correction to account for observational biases. These results add strength to the hypothesis that the binary fraction of CSGs is significantly lower than their main-sequence counterparts. Going forward, we stress the need for long-baseline multi-epoch spectroscopic surveys to cover the full parameter space of CSG binary systems.