Snake modules introduced by Mukhin and Young form a family of modules of quantum affine algebras. The aim of this paper is to prove that the Hernandez–Leclerc conjecture about monoidal categorifications of cluster algebras is true for prime snake modules of types Anand Bn. We prove that prime snake modules are real. We introduce S-systems consisting of equations satisfied by the q-characters of prime snake modules of types Anand Bn. Moreover, we show that every equation in the S-system of type An(respectively, Bn) corresponds to a mutation in the cluster algebra A(respectively, A) constructed by Hernandez and Leclerc and every prime snake module of type An(respectively, Bn) corresponds to some cluster variable in A(respectively, A). In particular, this proves that the Hernandez–Leclerc conjecture is true for all prime snake modules of types Anand Bn.