Given a holomorphic vector bundle $E$ on a compact complex Kähler manifold $X$ (I am happy to assume $X$ projective), we can compute the sheaf cohomology $H^ast(E)$ of $E$ using the Dolbeault complex with values in $E$ (see e.g. Huybrechts’ book Complex Geometry, Section 2.6).

The Dolbeault complex consists of the (smooth) global sections $Gamma(X, mathcal{A}^{0,q}_X otimes E)$ of the (smooth) bundle $mathcal{A}^{0,q}_X otimes E$. Is it known whether the (infinite-dimensional) vector space $Gamma(X, mathcal{A}^{0,q}_X otimes E)$ is generated by tensor products $omega otimes v$ with $omega$ respectively $v$ smooth sections of $mathcal{A}^{0,q}_X$ respectively $E$? I believe this does not need to hold.

Note that the description of the question Global Definition of the Dolbeault Complex of a Vector Bundle seems to assume this is true, whereas the comment underneath says that the operator $bar{partial}_E$ only locally agrees with $bar{partial} otimes id$ (which I interpret as a negative answer to my question).