Altermagnets are a new class of magnetic materials exhibiting strong time‐reversal symmetry‐breaking responses, akin to ferromagnets, alongside a fully compensated magnetization typical of collinear antiferromagnets (AFM). Altermagnets are predicted to be ideal platforms for THz spintronic devices due to their tunable chiral spin waves and robustness to stray magnetic fields. First principle calculations suggest that α‐MnTe (A‐type AFM order below TN = 310 K with the easy axis along the (210) direction) is an altermagnetic compound with a chiral split magnon spectrum with linear dispersion near the zone center. However, experimental evidence of an unconventional magnon spectrum is lacking. We propose to perform a circularly polarized momentum‐resolved Resonant Inelastic X‐ray Scattering at the L3 Mn edge below Tn to reveal the time‐reversal symmetry‐breaking nature of the low energy magnetic excitations in α‐MnTe.