I'm looking for certain (higher-dimensional) analogues of the Kodaira-Thurston manifolds, i.e. I want to know whether in $\dim_\mathbb{R}X\geq6$ we have examples of symplectic manifolds satisfying the following properties:
$X$ is a $\mathbb T^n$ bundle over $\mathbb T^n$, but not $\mathbb T^{2n}$.
The $\mathbb T^n$ bundle $\pi:X\rightarrow\mathbb T^n$ admits a section.
$X$ admits a complex structure.
Note that the work of Cordero-Fernandez-Gray (http://www.sciencedirect.com/science/article/pii/0040938386900509) provides some examples of symplectic manifolds which are $\mathbb T^{p+1}$ bundles over $\mathbb T^{p+1}$, which they call $M(p,0)$, and $M(1,0)$ gives you the Kodaira-Thurston manifold. These symplectic manifolds admit almost complex structures. Another candidate is the iwasawa manifold, but this is topologically a $\mathbb T^4$ bundle over $\mathbb T^2$.