Prions are self-replicative protein particles lacking nucleic acids. Originally discovered for causing infectious neurodegenerative disorders, they have also been found to play several physiological roles in a variety of species. Functional and pathogenic prions share a common mechanism of replication, characterized by the ability of an amyloid conformer to propagate by inducing the conversion of its physiological, soluble counterpart. Since time-resolved biophysical experiments are currently unable to provide full reconstruction of the physico-chemical mechanisms responsible for prion replication, one must rely on computer simulations. In this work, we show that a recently developed algorithm called Self-Consistent Path Sampling (SCPS) overcomes the computational limitations of plain MD and provides a viable tool to investigate prion replication processes using state-of-the-art all-atom force fields in explicit solvent. First, we validate the reliability of SCPS simulations by characterizing the folding of a class of small proteins and comparing against the results of plain MD simulations. Next, we use SCPS to investigate the replication of the prion forming domain of HET-s, a physiological fungal prion for which high-resolution structural data are available. Our atomistic reconstruction shows remarkable similarities with a previously reported mechanism of mammalian PrPSc propagation obtained using a simpler and more approximate path sampling algorithm. Together, these results suggest that the propagation of prions generated by evolutionary distant proteins may share common features. In particular, in both these cases, prions propagate their conformation through a very similar templating mechanism.