Real projective space $\mathbb{R}P^n$ embeds in a natural way in complex projective space $\mathbb{C}P^n$. (Using standard projective coordinates on $\mathbb{C}P^n$, $\mathbb{R}P^n$ is the subspace consisting of points that have a representative in which all coordinates are real.)

I know (very standard) cell decompositions for $\mathbb{C}P^n$ and $\mathbb{R}P^n$ that realize them as CW-complexes: $\mathbb{C}P^n$ has one cell in every even dimension while $\mathbb{R}P^n$ has one cell in every dimension. So in this standard CW-complex structure on $\mathbb{C}P^n$, $\mathbb{R}P^n$ does not occur as a subcomplex.

What I would like to know is this:

Is there a cell decomposition of $\mathbb{C}P^n$ that reveals $\mathbb{R}P^n$ as a subcomplex? What is it?

[An addition to the bounty notice. In case it makes a difference I will consider a non-trivial but non-general case as well, say $n=3$ or some such, for bounty. A general answer (affirmative or negative) is obviously preferred. Equally obviously, I'm not speaking for Ben. Have fun, JL]