Complex structure on product of two $n$-dimensional real manifolds 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?
 A: I'm going to take $M$ and $N$ to be connected. Note that they cannot have boundary. Furthermore, as complex manifolds are orientable, $M$ and $N$ must also be orientable.
If $n = 1$, then $M, N \in \{\mathbb{R}, S^1\}$. Note that $\mathbb{R}\times\mathbb{R}$ has a complex structure, in fact, precisely two: $\mathbb{D}$ and $\mathbb{C}$. In addition, $\mathbb{R}\times S^1 = S^1\times \mathbb{R}$ and $S^1\times S^1$ have uncountably many complex structures (annuli and tori respectively).
If $n = 2$, then $M$ and $N$ are orientable surfaces. These always admit complex structures, and therefore, so do their products.
Already for $n = 3$, we do not have a complete answer. In this case, the manifold $M\times N$ always admits an almost complex structure, but it is not clear when it admits an integrable one (i.e. a complex structure). This is a major open problem in complex geometry:

If a manifold admits an almost complex structure, does it admit a complex structure?

The answer is known to be false in real dimension four, but completely open in higher dimensions.
For $n = 4$, the manifold $M\times N$ may not even admit an almost complex structure. For example, $M = N = S^4$. Theorem 1 of this paper of Heaps completely characterises which closed eight-manifolds admit an almost complex structure. In principle, this could be used to determine what $M$ and $N$ could be, at least in the compact case.
A: 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:  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".
A: [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. 
