Assume $E\rightarrow X$ is a holomorphic vector bundle of rank $n$ with a linear connection $\nabla=\nabla^{1, 0}+\bar\partial_E$ which is compatible with the complex (somewhere the literature says "holomorphic") structure of $E$. Let $P\rightarrow X$ be the associated principal $GL_n(\mathbb{C})$-bundle (say the frame bundle of $E$).

There is a theorem which states that $\nabla$ induces a unique connection 1-form $\omega$ on $P$.

Let further $V$ be a representation of $GL_n(\mathbb{C})$ and let $P\times_{GL_n(\mathbb{C})}V$ be the associated vector bundle which should be in my opinion holomorphic as well. The principal connection $\omega$ induces always a linear connection $\tilde{\nabla}$ on the associated vector bundle $P\times_{GL_n(\mathbb{C})}V$. The question I want to pose is the following:

Is $\tilde{\nabla}$ compatible with the complex structure on $P\times_{GL_n(\mathbb{C})}V$?

P.S.:Thanks in advance for any clarifications, corrections and responses to that for me very interesting question.