Context. This study is aimed at identifying possible low-mass and sub-stellar companions to stars with well-determined metallicities. We investigate the multiplicity of metal-poor stars along with its impact on formation processes in the conditions of the early universe. Aims. Our goal is to look for wide common proper motion companions to metal-poor stars and study the binarity frequency at low metallicity with astrometry from large-scale catalogues, including Gaia, Visible and Infrared Survey Telescope for Astronomy (VISTA) Hemisphere Survey (VHS), and Wide Field Infrared Survey Explorer (WISE). Methods. We used the stellar parameter determination from the latest release of the Gaia catalogue to identify metal-poor stars over the entire sky. We combined the Gaia sample with other public catalogues and spectroscopic determinations for a given sub-sample to refine the stellar metallicities. We also considered, as input, other public catalogues of metal-poor stars to look for co-moving companions. We also obtained our own high-resolution images of a sub-sample with the lucky imaging technique. Results. We only found a few bona fide co-moving systems among a sample of 610 metal-poor stars with metallicities below ‑1.5 dex in the full sky. We inferred a multiplicity rate below 3%, with 3σ completeness for projected separations larger than 8 au, after taking into account incompleteness and any other limiting factors of our search. At closer separations, we found a minimum binary fraction of 20% that appears to be relatively independent of metallicity. Conclusions. We conclude that the multiplicity fraction of solar-type stars is relatively independent of metallicity for close-in companions with projected separations below ∼8 au. At separations between 8 and 10 000 au, the binary fraction of metal-poor stars drops significantly to a few percent and is significantly lower than the multiplicity derived for the solar-metallicity case. We interpret these similarities and differences as being due to the chemistry at work in molecular clouds as well as disruption effects attributed to the old age of sub-dwarfs.