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Astrometric noise (AEN) in excess of parallax and proper motion is a potential signature of orbital wobble of individual components in binary star systems. The combination of X-ray selection with astrometric noise could then be a powerful tool for robustly isolating accreting binaries in large surveys. Here, we mine the Gaia EDR3 catalogue for Galactic sources with significant values of astrometric noise over the parameter space expected for known and candidate X-ray binaries (XRBs). Cross-matching our sample with the Chandra Source Catalogue returns a primary sample of ~6,500 X-ray sources with significant AEN. X-ray detection efficiency for objects with significant AEN is a factor of ~4.5 times higher than in a matched control sample exhibiting low AEN. The primary sample branches off the main sequence much more than control objects in colour-mag space, and includes a higher fraction of known binaries, variables and young stellar object class types. However, values of AEN reported in the Gaia pipeline releases so far can exceed expectations for individual XRBs with known semi-major axis size and other system parameters. It is likely that other factors (possibly attitude and modelling uncertainties, as well as source variability) currently dominate the observed excess noise in such systems. Confirmation of their nature must therefore await future Gaia releases. The full X-ray matched catalogue is released here to enable legacy follow-up.