Let $V,V'\subset T\mathbf{CP}^3$ be smooth (real) 2-dimensional subbundles of the tangent bundle of complex projective 3-space. Suppose that $V$ and $V'$ are isomorphic as topological vector bundles, i.e. there exists a continuous vector bundle isomorphism between $V$ and $V'$.
Are $V$ and $V'$ necessarily isomorphic as smooth vector bundles (i.e. does there exist a smooth map $V\to V'$ which induces a linear isomorphism on the fibers)?
I am aware of a classification of complex vector bundles over projective spaces, but couldn't find anything on real smooth vector bundles. I also think that the fact that both are subbundles of the tangent bundle should make this easy, but I don't know how to use that condition.