Existence of Complex Frames on a Complex Vector Bundle $E \rightarrow M $ be a complex vector bundle (of real rank $2r$) with almost complex structure
$J:E\rightarrow E \space\space\space(J^2 =-1)$ on it. $U\subset M$ be a trivial neighbourhood.
Does there exist a complex frame for $E \rightarrow M$ over U? By a complex frame, I mean a set of $r$ (real) linearly independent $E_U$-sections $\{X_1, ... , X_r\}$, such that $\{X_1, ... , X_r, JX_1, ... , JX_r\}$ forms a complete $E_U$-frame. (By "linearly independent", I mean they are linearly independent in the fibre above every point $x\in U$). It seems like such frames do exist as I recall seeing such frames in definition of complex connection and curvature matrices on complex vector bundles, which then lead to construction of Chern forms and classes.
I need such a frame to prove that existence of a $J$ map for $E \rightarrow M $ implies a reduction of structure group of E to $GL(r,C)$. I am stuck with this, even though I can prove the converse by explicitly constructing a complex frame using the reduction.
Edit: I am using the following definition for 'complex vector bundle': A real even ranked vector bundle with a J-map. Sorry for the confusion.
 A: The definition of complex vector bundle you're using is non-standard. The standard definition mirrors the definition of a real vector bundle, but with $\mathbb{R}$ replaced by $\mathbb{C}$. Your definition is equivalent though, see this question.
To the question at hand: if $U \subset M$ is such that $E|_U$ is trivial as a real vector bundle, is it necessarily trivial as a complex vector bundle? The answer is no.
Example: Consider the complex vector bundle $E = TS^2\oplus\varepsilon_{\mathbb{C}}^1$ over $S^2$. The underlying real vector bundle is $TS^2\oplus\varepsilon_{\mathbb{R}}^2 \cong \varepsilon_{\mathbb{R}}^4$, i.e. it is trivial. On the other hand, the first Chern class of the complex vector bundle $E$ is $c_1(E) = c_1(TS^2\oplus\varepsilon_{\mathbb{C}}^1) = c_1(TS^2) \neq 0$, so it is not trivial. Therefore, by taking $U = S^2$, we have an example where $E|_U$ is trivial as a real vector bundle, but not as a complex vector bundle.
Regarding the reduction of structure group, it is true that if $E|_U$ is trivial as a real bundle, then there is $U' \subseteq U$ such that $E|_{U'}$ is trivial as a complex vector bundle, see the answer to the aforementioned question.
