I apologize for that the following answer may result from a total misunderstanding of the problem. .
Let $X=M\times N$, and let $E=\pi_M^*TM$ and $F=\pi_N^*TN$. Note that $TX=E\oplus F$. As suggested in Ben McKay's answer, you could first look for the existence of an almost complex structure $J\in End(TX)$, $J^2=-Id$, for which the eigen-bundles of $\pm i$ are precisely $E^{\mathbb{C}}$ and $F^{\mathbb{C}}$.
This may exist or not. It does for instance under the assumption that there exists an isomorphism of vector bundles $\phi:E\to F$. In that case, define $$ J:=\left(\matrix{0 &\phi^{-1} \\ -\phi & 0}\right) $$ In this situation it seems to me that the almost complex structure is automatically integrable: indeed, sections of $E^{\mathbb{C}}=\pi_M^*TM^{\mathbb{C}}$ are stable under the Lie bracket.