dg.differential geometry – Pull back of Spin$^{mathbb{C}}$ bundle

Let $M$ be a closed $4$-d Riemannian manifold and $Z$ be its twistor space of $M$, i.e., the bundle of almost complex structures on $M$. Let $V$ be a Spin$^{mathbb{C}}$ bundle, $V_+$ denote the positive spin bundle. Now we know $Z$ admits more than one almost complex structure. So it can have canonical Spin$^{mathbb{C}}$ bundle once we fix an almost complex structure and we can have other Spin$^{mathbb{C}}$ bundles by twisting it with a complex line bundle. Now let $pi:Zrightarrow M$ be the projection. Can we realize $pi^*(V_+)$ as a sub-bundle of some Spin$^{mathbb{C}}$ bundle of positive spinors on $Z$ and if yes how?