Consider the product of $2n$ two-spheres $X_n=(S^2)^{2n}$. This manifold admits an orientation preserving involution that preserves the product structure and acts as the (orientation reversing) central symmetry for each $S^2$. Is it possible to say for which $n$ the quotient manifold $X_n/\mathbb Z_2$ admits an almost complex structure? I believe that if such $n$ exists then $n>1$, i.e. $S^2\times S^2/\mathbb Z_2$ is not almost complex (for the defined action).