Blue straggler candidates in Haffner 10: testing a Gaia colour–astrometry diagnostic against data-quality systematics

While long-term radial velocity (RV) monitoring has successfully mapped blue straggler star (BSS) multiplicity in nearby, unreddened open clusters, such spectroscopic surveys are observationally prohibitive for distant ($d \gt 2$ kpc) and moderately reddened Galactic clusters. To bypass this limitation, we present a general statistical framework that combines an empirical colour–magnitude diagram (CMD) displacement parameter ($\eta _{\rm binary}$) with Gaia astrometric noise (renormalized unit-weight error, RUWE) to diagnose BSS multiplicity on a population level. We apply this methodology to the intermediate-age open cluster Haffner 10 ($d \simeq 3.5$ kpc, $A_V \simeq 1.2$ mag) as a representative test bed. Out of 52 identified BSS candidates, the population is strongly redward-dominated (47 stars, $f_{\rm red} \simeq 0.90$). After correcting for differential reddening, we find a non-negligible residual main-sequence colour dispersion ($\sigma _{\Delta C} = 0.105$ mag), showing that individual CMD positions are insufficient to identify formation channels. However, a Spearman rank test reveals a positive correlation between the continuous CMD displacement $\eta _{\rm binary}$ and RUWE ($\rho _{\rm S}=0.35$, $p=0.011$), indicating that redward-displaced BSS candidates tend to show enhanced astrometric residuals. We compare our results with a literature sample of well-studied open clusters (M67, NGC 188, Melotte 66, and NGC 7789) to discuss how observational limitations and dynamical states shape the observed BSS configurations. Our approach offers a highly cost-effective template for characterizing BSS origins in distant open clusters where spectroscopy is currently unavailable.