Let $F\to M$ be a holomorphic vector bundle over a complex manifold $M$ and let $s:M\to F$ be a no-zero section. Let $E$ be the complexification of $F$, and suppose that $E$ admits a holomorphic structure (the transition maps need not be the ones arising from $F$). Furthermore, let $\nabla$ be the Chern connection of $E$.
We'll say that $s$ is the real part of a holomorphic section if there is a holomorphic section $v$ and a real section $t$ such that, in a trivialization, we can express $$v = s+it$$
My question: is there a coordinate free way to express this condition? That is, a coordinate free manifestation of the Cauchy-Riemann equations?
In some special situations, it is possible to do so in coordinates. Or if $F=TM$, and $J$ is the almost complex structure on $M$, then one just needs to check if the (1,0)-component $s-iJs$ is holomorphic. I'm not sure about a general vector bundle. I should also remark I'm mostly interested in the case where $M$ is a Riemann surface.
