I'm trying to solve the following problem related to characteristic classes:
Is it possible to find a complex vector bundle on $S^2$ with nonzero first Chern class, which is trivial as a real vector bundle?
I suspect the answer is no, and I'm trying to prove that. If we have such a bundle $E$, then we know that $w_2(E)$ is the image of $c_1(E)$ under the coefficient homomorphism $H^2(S^2;\Bbb Z)\to H^2(S^2;\Bbb Z/2\Bbb Z)$. I would like to show that $c_1(E)\neq0$ implies $w_2(E)\neq0$, so that $E$ cannot be trivial as a real vector bundle, but I am unsure how to do this. Does anybody have any suggestions?
First of all, $c_1(E) \neq 0$ does not imply $w_2(E) \neq 0$. For example, thinking of $S^2$ as $\mathbb{CP}^1$, $TS^2$ is a complex vector bundle and $c_1(TS^2) = e(TS^2) \neq 0$ (in fact, $\langle e(TS^2), [S^2]\rangle = \chi(S^2) = 2$). However $w_2(TS^2) = w_2(TS^2\oplus\varepsilon^1) = w_2(\varepsilon^3) = 0$.
If $\operatorname{rank}_{\mathbb{C}} E = 1$, then the answer to your question is no.
Let $\sigma$ be a nowhere zero section of $E$. Then there is a bundle isomorphism $E \to S^2\times\mathbb{C}$ given by $v_x \mapsto (x, \lambda_v)$ where $\lambda_v \in \mathbb{C}$ is the unique complex number satisfying $v_x = \lambda_v\sigma(x)$. So $E$ is trivial and therefore $c_1(E) = 0$.
Note, the above argument works for any base. In general, a complex line bundle is trivial if and only if the underlying real rank two bundle is trivial. This is no longer true for holomorphic line bundles though, i.e. there are non-trivial holomorphic line bundles which are trivial as complex line bundles, see here for example.
For $\operatorname{rank}_{\mathbb{C}} E > 1$, the answer to the question you posed is yes.
Consider the rank two complex vector bundle $E = TS^2\oplus\varepsilon^1_{\mathbb{C}}$. Note that $c_1(E) = c_1(TS^2) \neq 0$; in particular, $E$ is not trivial as a complex vector bundle. As a real rank four bundle though, $E = TS^2\oplus\varepsilon^2_{\mathbb{R}} = \varepsilon^4_{\mathbb{R}}$, i.e. $E$ is trivial. By direct summing with $\varepsilon^k_{\mathbb{C}}$ instead of $\varepsilon^1_{\mathbb{C}}$, we obtain examples for any complex rank greater than one.
The reason such an example exists is because $TS^2$ is a stably trivial real bundle, but it is not a stably trivial complex bundle. More generally, for $g \neq 1$, $E = T\Sigma_g\oplus\varepsilon^1_{\mathbb{C}}$ has non-zero first Chern class but is trivial as a real bundle.