In the so-called “alpha-heating” experiment performed on the JET tokamak during the deuterium-tritium campaign of 1997, the ion temperature was found to be far exceeding (both in absolute value and in its rise time) the level that could have been expected from direct collisional heating by the fusion-born alpha particles themselves and energy equi-partition with the electrons. To date, no explanation has been put forward for this long standing puzzle, despite much work having been performed on this subject in the early 2000s. Two analysis methods that have recently become available have been employed to re-analyse these observations of an anomalous ion heating. First, an algorithm based on the Sparse Representation of Signals has been used to analyse magnetic, reflectometry and electron cyclotron emission measurements of the turbulence spectra in the drift-wave range of frequencies. This analysis has then been complemented with turbulence simulations performed with the GENE code. We find, both experimentally and in the simulations, that the presence of a minority, but sufficiently large, population of fusion-born alpha particles that have not yet fully thermalized stabilizes the turbulence in the ion-drift direction, but practically does not affect the turbulence in the electron-drift direction. We link such stabilization of the ion-drift-wave turbulence to the increase in the ion temperature above the level achieved in similar discharges that did not have (at all or enough) alpha particles. When the fusion-born alpha particles have fully thermalized, the turbulence spectrum in the ion-drift direction re-appears at somewhat larger amplitudes, which we link to the ensuing reduction in the ion temperature. This phenomenological dynamics fully corresponds to the actual experimental observations. By taking into account an effect of the alpha particles that had not been previously considered, our new analysis finally presents a phenomenological explanation for the so-far-unexplained anomalous ion heating observed in the JET alpha-heating experiment of 1997. Through the formulation of an empirical criterion for ion-drift-wave turbulence stabilization by fusion-born alpha particles, we also show why similar observations were not made in the other deuterium-tritium experiments run so far in JET and TFTR. This allows assessing the operational domain for this stabilization mechanism for ion-drift-wave turbulence in future burning plasma experiments such as ITER, which may open a new path towards the sustainment of a high energy gain in such forthcoming devices.