Existence of a real structure on the tangent bundle of a complex manifold

Question

Let $X$ be a compact complex manifold. What are obstructions to the existence of a real subbundle $V$ of $TX$ such that $TX = V \otimes \mathbb{C}$?

For example, does $\mathbb{CP}^n$ have such a structure?

Answer 1

For a vector bundle to have real structure, it must be isomorphic to its complex conjugate, hence it's $i$th Chern class must be equal to its own negation (i.e. $2$-torsion) for all odd $i$. So $2c_1, 2c_3, 2 c_5, \dots$ are all obstructions.

Since $2c_1$ is nonzero for the tangent bundle of projective space, $\mathbb C\mathbb P^n$ has no such structure.