I'm trying to understand the proof Proposition 2.3 in this paper. Specifically, I'm having difficulty understanding the last paragraph where they claim that
The given Chern connection $\nabla$ for $E$ on $\mathcal{X}_t$ defines a holomorphic structure in the induced projective bundle $\mathbb{P}(E)$ with respect to another complex structure $s\in \mathbb{P}(\omega)$ if and only if the tracefree part $F_0$ of $F$ [...] is type (1,1) with respect to $s$.
The setup is $X$ is a K3 surface with Kahler form $\omega$, and $\mathcal{X}$ is the twistor family for $X$, which is a bundle $\mathcal{X}\to \mathbb{P}(\omega)\cong \mathbb{P}^1$ where the fiber over a point $t=(a,b,c)\in \mathbb{P}(\omega)$ is the underlying manifold of $X$ equipped with the complex structure determined by $\omega_t = a\omega_I + b\omega_J + c\omega_K$. $E$ is assumed to be a $\omega_t$-stable holomorphic hermitian vector bundle over $\mathcal{X}_t$.
Question: How does one prove this statement?
I think I'm mainly confused on how I can relate $F_0$ to the existence of a complex structure on $\mathbb{P}(E)$, and the comments in the paper about this statement were of little help to me. They say to repeat their argument when $\det(E)$ is trivial with $F_0$ in place of $F$, but it seems to rely on the fact that a complex structure on a vector bundle is determined by a $\overline{\partial}$-operator, which is a statement that (I don't think) holds true for projectivized bundles.