We use deep, high quality colour magnitude diagrams obtained with the Hubble Space Telescope to compute a simplified version of the Mironov index [SMI; B/(B+R)] to parametrize the horizontal branch (HB) morphology for 23 globular clusters in the M31 galaxy (Sample-A), all located in the outer halo at projected distances between 10 kpc and 100 kpc. This allows us to compare them with their Galactic counterparts, for which we estimated the SMI exactly in the same way, in the SMI vs. [Fe/H] plane. We find that the majority of the considered M31 clusters lie in a significantly different locus, in this plane, with respect to Galactic clusters lying at any distance from the center of the Milky Way. In particular they have redder HB morphologies at a given metallicity, or, in other words, clusters with the same SMI value are ~0.4 dex more metal rich in the Milky Way than in M31. We discuss the possible origin of this difference and we conclude that the most likely explanation is that many globular clusters in the outer halo of M31 formed ~1-2 Gyr later than their counterparts in the outer halo of the Milky Way, while differences in the cluster-to-cluster distribution of He abundance of individual stars may also play a role. The analysis of another sample of 25 bright M31 clusters (eighteen of them with M_V<= -9.0, Sample-B), whose SMI estimates are much more uncertain as they are computed on shallow colour magnitude diagrams, suggests that extended blue HB tails can be relatively frequent among the most massive M31 globular clusters, possibly hinting at the presence of multiple populations.
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