If you have a Hilbert space $\mathcal{H}$, you can make a projective Hilbert space by modding out by the $U(1)$ group action of multiplying by a phase, $\mathcal{H}/U(1)$.

While we usually talk about states as vectors in a Hilbert space, really they are rays in a projective Hilbert space because phases are unobservable. This has some physical consequences like the existence of half integer spin.

However, there seems to me to be something you cannot do in a projective Hilbert space: add states. While there is no difference between $|\psi\rangle$ and $-|\psi\rangle$, there is a big difference between $|\phi\rangle + |\psi\rangle$ and $|\phi\rangle - |\psi\rangle$. So how can states be added together in a self consistent way in a projective Hilbert space?