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After doing a bit of playing around (I think) I was able to show that the map $\operatorname{id}\otimes\ \psi : \Omega^{p,q}(X, E) \to \Omega^{p,q}(X, \bar{E})$, where $\psi $ is the conjugation map $E \to \bar{E}$, is well-defined for a complex vector bundle $E$ provided that $E$ has real transition functions. The trivial complex bundle has this property, but are there others? If so, what can we say about them?


Added later: Alex has given some examples in the comments below so I am left with the following question.

Can we classify complex vector bundles with real transition functions?

Classify may be to strong a requirement. What I am after is either a name for this class of bundles or some properties which demonstrate how few there are. Compare this to the constant transition functions scenario in which case the bundles are flat and we can extend the exterior derivative to $\Omega^{\bullet}(X, E)$ by $d\otimes\operatorname{id}_E$.

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The trivial bundle? –  Alex Youcis Dec 10 '12 at 6:38
    
Of course. I'll adjust my question. –  Michael Albanese Dec 10 '12 at 6:46
    
I'm still a little confused. Are you looking for smooth complex vector bundles or holomorphic? If you are looking for smooth a very obvious way to make real-valued transition functions is by taking any, say, line bundle $L$ and considering $L\otimes \overline{L}$. This will give you a line bundle whose transition functions are the modulus squared of the transition functions of $L$. –  Alex Youcis Dec 10 '12 at 13:03
    
Smooth. I suppose I want to know whether such bundles can be classified. –  Michael Albanese Dec 11 '12 at 11:56
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