Let $M$, $N$ be $n$-dimensional real manifolds. Does $M\times N$ admits a complex structure? If not, are there known condidtions ensuring that $M\times N$ admits a complex structure?
['https://mathoverflow.net/questions/346122', 'https://mathoverflow.net', 'https://mathoverflow.net/users/147073/']
I assume you are asking if $M \times N$ admits a complex structure.
I believe that there is no result known on this question, although we might have something to say about the existence of an almost complex structure for which both factors are totally real.
Once again we can quote [Gromov](https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/SpacesandQuestions.pdf): Page 30. "How much do we gain in global understanding of a compact $(V, J)$ by assuming that the structure $J$ is integrable (i.e. complex)? It seems nothing at all: there is no single result concerning all compact complex manifolds".
[EDIT: I started to write this answer, and then realized it doesn't really answers the question - I leave it here anyway, just in case]

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 both projections $\pi\_M$ and $\pi\_N$ have totally real fibers. 

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 the eigenbundle $T^{1,0}X$ corresponding to $+i$ is isomorphic to $E^{\mathbb{C}}$ in the following way: 
$$
E^{\mathbb{C}}\to T^{1,0}X\quad;\quad v\mapsto v+i\phi(v)
$$
You may want to look for conditions on $\phi$ for $J$ to be integrable.