This seems to be a very natural problems, but I have not found any mention of it anywhere.
Let $\mathbb{R}^n \subset \mathbb{C}^n$; $V$ — $k$-dimensional complex vector subspace of $\mathbb{C}^n$. What real dimension can $V \cap \mathbb{R}^n$ have?
The only condition I have found is the obvious one: if $k > n/2$, $V$ and $\mathbb{R}^n$ must have a non-zero intersection by dimension count.
Since $$ \dim_{\mathbb R}(\mathbb C^n)\ge\dim_{\mathbb R}(V+\mathbb R^n)=\dim_{\mathbb R}(V)+\dim_{\mathbb R}(\mathbb R^n)-\dim_{\mathbb R}(V\cap\mathbb R^n), $$ we have $\dim_{\mathbb R}(V\cap\mathbb R^n)\ge 2k+n-2n=2k-n$. Consequently, $$ n\ge\dim_{\mathbb R}(V\cap\mathbb R^n)\ge\max(2k-n,0). $$