dg.differential geometry – almost complex $mathbb{Z}^{6}$-action

Suppose we take an almost complex structure on $mathbb{T}^{6}$ with $c_{1} neq 0$ (there should be infinitely many homotopy classes satisfying this requirement). Now pull it back to the universal cover $mathbb{R}^6$, giving an almost complex structure $J$, it should be invariant by the deck group action of $mathbb{Z}^{6}$.

My question is there a case where we can explicitly describe $J$ and the group action of $mathbb{Z}^6$ in coordinates?

(I asked this question in stack exchange in August https://math.stackexchange.com/questions/3789163/almost-complex-actions-on-mathbbr6 with no answer so I thought I would ask here.)