Given the uncertain evolutionary status of blue supergiant stars, their multiplicity properties hold vital clues to better understand their origin and evolution. As part of The Binarity at LOw Metallicity (BLOeM) campaign in the Small Magellanic Cloud, we present a multi-epoch spectroscopic survey of 128 supergiant stars of spectral type B5–F5, which roughly correspond to initial masses in the 6–30 M⊙ range. The observed binary fraction for the B5–9 supergiants is 25 ± 6% (10 ± 4%) and 5 ± 2% (0%) for the A–F stars, which were found using a radial-velocity (RV) variability threshold of 5 km s‑1 (10 km s‑1) as a criterion for binarity. Accounting for observational biases, we find an intrinsic multiplicity fraction of less than 18% for the B5–9 stars and 8‑7+9% for the AF stars, for the orbital periods up to 103.5 days and mass ratios (q) in the 0.1 < q < 1 range. The large stellar radii of these supergiant stars prevent short orbital periods, but we demonstrate that this effect alone cannot explain our results. We assessed the spectra and RV time series of the detected binary systems and find that only a small fraction display convincing solutions. We conclude that the multiplicity fractions are compromised by intrinsic stellar variability, such that the true multiplicity fraction may be significantly smaller. Our main conclusions from comparing the multiplicity properties of the B5–9- and AF-type supergiants to that of their less evolved counterparts is that such stars cannot be explained by a direct evolution from the main sequence. Furthermore, by comparing their multiplicity properties to red supergiant stars, we conclude that the AF supergiant stars are neither progenitors nor descendants of red supergiants.