at.algebraic topology – Even, non liftable Stiefel-Whitney class

Let $M$ be a smooth manifold and $E$ a smooth real vector bundle of even rank over $M$.
If $E$ admits of a complex vector bundle structure $mathcal E$ ($mathcal E_mathbb R=E$) then all odd Stiefel-Whitney classes of $E$ vanish: $$w_{2i+1}(E)=0$$
Moreover the even Stiefel-Whitney classes of $E$ are the images under the reduction morphism $operatorname {red}^{2i}:H^{2i}(M,mathbb Z)to H^{2i}(M,mathbb F_2)$ of its Chern classes, namely $$w_{2i}(E)=operatorname {red}^{2i}(c_i(mathcal E))$$
My question
Is there a real vector bundle of even rank $E$ with all odd $w_{2i+1}(E)=0$ that nevertheless cannot be endowed with a complex structure just because some $w_{2i}(E)in H^{2i}(M,mathbb F_2)$ cannot be lifted to $mathbb Z$?
Explicitly, the equation $$operatorname {red}^{2i}(c_i)=w_{2i}(E)in H^{2i}(M,mathbb F_2)$$ has no solution $c_iin H^{2i}(M,mathbb Z)$ .
(The answer is probably quite easy but not for me classical algebraic geometer very unexperienced with real vector bundles.)