¹²C/¹³C as a function of log(C/O) for core He-burning and RGB stars in our sample (orange squares and circles, respectively) compared to the BGM simulations for core He-burning stars only. The simulations (grey dots) take into account the thermohaline mixing (middle and right panels) or not (left panel). The C/O ratio derived by Blanton et al. (2017) for stars in common is also shown, using our carbon isotopic ratios (blue stars in middle panel). The abundances derived by Gałan et al. (2017, 2023), and Gratton et al. (2000) are also compared with the simulation in the right panel. The ¹²C/¹³C at the core He-burning phase predicted by stellar evolution models computed by Lagarde et al. (2012a) and including both thermohaline and rotation-induced mixing (at V/V_crit ∼ 0.3) are shown at [Fe/H] = 0 and [Fe/H] = −0.86 (green dotted and solid lines, respectively).