  We construct a family of $6$-dimensional compact manifolds $M(A)$, which are
simultaneously diffeomorphic to complex Calabi-Yau manifolds and symplectic
Calabi-Yau manifolds. They have fundamental groups $\mathbb{Z} \oplus
\mathbb{Z}$, their odd-dimensional Betti numbers are even, they satisfy the
hard Lefschetz property, and their real homotopy types are formal. However,
$M(A) \times Y$ are not homotopy equivalent to any compact K\"ahler manifold
for any topological space $Y$.
