Let $M$ be a manifold of even dimension and let $\pi:\mathcal J\rightarrow M$ be its twistor bundle, which has as its fibre at $p\in M$, the space of endomorphisms $J_p\in \mathrm{End}(T_pM)$ satisfying $J_p^2 =-\mathrm{id}_{T_pM}$. Since there is a transitive action of $\mathrm{GL}(T_pM)$ on this space via conjugation, it can be described as a coset space $\mathrm{GL}(T_pM)/\mathrm{GL}(T_pM,J_p)$, where $\mathrm{GL}(T_pM,J_p)$ denotes the centraliser of $J_p$.
Sections of the twistor bundle $\mathcal J$ correspond to almost complex structures on $M$. Is there a nice way to characterise when such a section corresponds to an integrable complex structure in terms of how it sits inside the twistor bundle $\mathcal J$?
I know that having an integrable complex structure is equivalent to the existence of a (linear) torsion-free connection on $M$ with respect to which the almost complex structure is parallel, and that linear connections on $M$ induce horizontal distributions (i.e. Ehresmann connections) on $\mathcal J$, but I can't find a good characterisation of the torsion-free condition in terms of the induced Ehresmann connection on $\mathcal J$.
As a bonus question, is there also a good way to characterise when a section of $\mathcal J$ admits a compatible Kähler metric in terms of how the section sits inside $\mathcal J$?