Let X be an irreducible Hermitian symmetric space of non-compact type of dimension greater than 1 and G be the group of biholomorphisms of X ; let be a quotient of X by a torsion-free discrete subgroup of G such that M is of finite volume in the canonical metric. Then, due to the G-equivariant Borel embedding of X into its compact dual Xc, the locally symmetric structure of M can be considered as a special kind of a -structure on M, a maximal atlas of Xc-valued charts with locally constant transition maps in the complexified group . By Mostow's rigidity theorem the locally symmetric structure of M is unique. We prove that the -structure of M is the unique one compatible with its complex structure. In the rank one case this result is due to Mok and Yeung.