Principal circle bundle on real projective three-space (${\mathbb{RP}}^3⟶{\mathbb{CP}}^1$)

Write a point of Real projective space as a nonzero vector $(w_1,w_2)\in {\mathbb{C}}^2$ modulo real nonzero scaling. The map to Complex projective space sends its class to the complex line through that vector. A free right action of the circle group is $[w]\cdot u=[\lambda w]$ where ${\lambda }^2=u$; the two possible roots differ by a real sign and therefore give the same projective point. Local smooth square-root branches prove smoothness. Over $[z:1]$, the fibre coordinate is $(w_2/|w_2|{)}^2$; over $[1:\zeta ]$, it is $(w_1/|w_1|{)}^2$. On the overlap $\zeta =1/z$, the transition function of a principal bundle is $u\mapsto (z/\overline{z})u$. Its phase winds twice. Ordinary scalar multiplication by $u$ instead has a kernel of order two and is not a free action of a group.