In a recent work in collaboration with Chiara Camere and Alessandra Sarti, we introduce logarithmic Enriques varieties as a singular analogue of Enriques manifolds, generalizing the notion of log-Enriques surfaces introduced by Zhang in 1991. In this talk, I will mainly focus on the possible torsion indices of the subfamily of logarithmic Enriques varieties that admit a quasi-étale cover by a singular symplectic variety, and I will give some examples.