Twistor space, as complex projective space $\mathbb{CP}^3$, is related to Minkowski 4-D space time (metric $1, -1,-1,-1$), by the incidence relation.
Let $Z = (v_a, u^{\dot{a}})$ a point in $\mathbb{CP}^3$, where $v_a$ and $u^{\dot{a}}$ are 2-complex components spinors.
Let $x_{a\dot{a}} = \sum x^\mu (\sigma_\mu)_{a\dot{a}}$, where $\sigma_\mu$ are the Pauli matrices, and $x^\mu$ a point in Minkowski 4-D space-time.
The incidence relation is then $v_a = x_{a\dot{a}} u^{\dot{a}}$
The representation of a space-time point, in the twistor space $\mathbb{CP}^3$, is a complex line $\mathbb{CP}^1$.
So, $\mathbb{CP}^3$ may be seen as a bundle space, with base Minkowski space-time, and fiber $\mathbb{CP}^1$.
But, is the inverse true, that is: could we see $\mathbb{CP}^3$ as a bundle space, with base $\mathbb{CP}^1$, and as fiber the 4-D Minkowski space-time?