underlying real vector bundle of a complex vector bundle

Question

Let $\eta^\mathbb{C}$ be a complex line bundle. If the underlying $2$-dimensional vector bundle $\eta$ is not trivial as a real vector bundle, can we obtain that $\eta^\mathbb{C}$ is not trivial as a complex line bundle?

I get confused. Could you give a valid proof? thanks so much!

Answer 1

Otherwise a complex trivialization would give you also a real one

$$ E(\eta) \cong B(\eta)\times \mathbb C^n \cong B(\eta ) \times \mathbb R^{2n}. $$