In this paper, discussing the Hopf fibration, the autors state:
Its non trivial character implies that $S_3 \neq S_2 \times S_1$. This non trivial character translates here into the known failure in ascribing consistantly a definite phase to each representing point on the Bloch sphere.
I'm struggling to understand the connection made in the second sentence.
Related question: Why do we need non-trivial fibrations?
If you could ascribe a non-trivial phase $\theta$ to each point on the Bloch sphere, that would be a function $\theta : S^2 \to S^1$. That would mean the $S^1$-bundle in which the phase lives is the trivial bundle $S^2\times S^1$ - a principal bundle with a global section is trivial, this is perhaps the mathematical fact you are missing to interpret the quoted section.
But you know that the correct space of "states with phases" is $S^3$ and since $S^3\neq S^2\times S^1$ it is not trivial and possesses no global section, hence $\theta$ cannot exist.
