Vector bundle over an oriented manifold with non-vanishing w_2w_3

As far as I know the Wu manifold $X=SU(3)/SO(3)$ is orientable and has mod 2 cohomology ring $H^*(X;\mathbb{Z}_2)=\Lambda(\omega_2(X),\omega_3(X))$. Thus $\omega_2(X)\cdot\omega_3(X)\neq 0$, and in fact generates $H^5(X;\mathbb{Z}_2)$.

I think you can take the manifold to be $M = \mathbb{RP}^5$ and the vector bundle $V$ to be the direct sum of $3$ copies of the tautological line bundle $\gamma$ on $M$. Let $x = w_1(\gamma)\in H^1(M,\mathbb{Z}/2\mathbb{Z})$. Then by the Whitney product formula you get $w_2(V) = x^2$ and $w_3(V) = x^3$, so $w_2(V)w_3(V) = x^5$. But this is non-zero in the mod 2 cohomology of $\mathbb{RP}^5$, which is, as an algebra, isomorphic to $\mathbb{Z}/2\mathbb{Z}[x]/(x^6)$.