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We construct a geometric model of eight-dimensional manifolds and realize them in the context of type II string theory. These eight-manifolds are constructed by non-trivial T4 fibrations over Calabi–Yau two-folds. These give rise to eight-dimensional non-Kähler Hermitian manifolds with SU(4) structure. The eight-manifold is also a circle fibration over a seven-dimensional G2 manifold with skew torsion. The eight-manifolds of this type appear as internal manifolds with SU(4) structure in type IIB string theory with F3 and F7 fluxes. These manifolds have generalized calibrated cycles in the presence of fluxes.