Real structure on complex vector bundle In Chapter 19 of Homotopical Topology by Fomenko and Fuchs, there is an exercise asserting that a complex vector bundle $\xi$ is the complexification of a real vector bundle if and only if there is an isomorphism $\xi\cong\bar\xi$. I have been trying to do this exercise, but getting caught up on some of the details.
I know that if $V$ is a complex vector space, then specifying a real structure on $V$ (i.e. a subspace $W\subset V$ with $V=W\oplus iW$) is equivalent to specifying a conjugate-linear involution on $V$. However, I have been unable to go from a general conjugate-linear isomorphism $\xi\rightarrow\xi$ to one that equals its own inverse.
Any suggestions?
 A: It turns out that the exercise is wrong. The following answer is based on a homework problem assigned by Alexander Givental and multiple conversations with Tong Zhou.
For the quaternionic projective plane $\mathbb H\mathbb P^1$, we take the set of right-lines in $\mathbb H^2$ (i.e. let $\mathbb H^\times$ act on $\mathbb H^2$ by right-multiplication and define $\mathbb H\mathbb P^1$ to be the quotient of $\mathbb H^2-\{0\}$ under this action). Every $\ell\in \mathbb H\mathbb P^1$ is a quaternionic line under right-multiplication. This makes the tautological bundle $\xi=\{(\ell,v)\in \mathbb H\mathbb P^1\times \mathbb H^2\mid v\in \ell\}$ into a quaternionic line bundle. We can forget the quaternionic structure and consider the complex vector bundle $\xi|\mathbb C$ over $\mathbb H\mathbb P^1$. Right-multiplication by $j$ preserves each fiber and anti-commutes with $i$, so we get a conjugate-linear automorphism of $\xi|\mathbb C$.
However, the quaternionic line bundle $\xi$ is non-trivial, so it has no non-vanishing sections. But if $\eta$ is any real vector bundle over $S^4$ of dimension 2, then the Euler class $e(\eta)\in H^2(S^4)=0$ vanishes, so $\eta$ admits a non-vanishing section. Therefore, the complexified bundle $\eta\otimes \mathbb C$ admits a section as well. Since $\mathbb H\mathbb P^1\cong S^4$, no real bundle $\eta$ over $\mathbb H\mathbb P^1$ can satisfy $\eta\otimes \mathbb C\cong \xi|\mathbb C$.
