Explosive crystallization is a well known phenomenon occurring due to the thermodynamic instability of strongly under-cooled liquids, which is particularly relevant in pulsed laser annealing processes of amorphous semiconductor materials due to the globally exothermic amorphous-to-liquid-to-crystal transition pathway. In spite of the assessed understanding of this phenomenon, quantitative predictions of the material kinetics promoted by explosive crystallization are hardly achieved due to the lack of a consistent model able to simulate the concurrent kinetics of the amorphous-liquid and liquid-crystal interfaces. Here, we propose a multi-well phase-field model specifically suited for the simulation of explosive crystallization induced by pulsed laser irradiation in the nanosecond time scale. The numerical implementation of the model is robust despite the dis-continuous jumps of the interface speed induced by the phenomenon. The predictive potential of the simulations is demonstrated by means of comparisons of the modelling predictions with experimental data in terms of in situ reflectivity measurements and ex-situ micro-structural and chemical characterization.