I understand that the top Stiefel Whitney class is an obstruction for the tangent bundle of a manifold to have a trivial line sub-bundle. I am looking for a counterexample when removing the word "trivial", i.e: A compact manifold $M$ of dimention n such that $w_n(TM)\neq0$ (or equivalently $\chi(M)$ is odd) and there exists a sub-bundle $\xi\subset TM$ with $\operatorname{rank}(\xi)=1$.

After some thought I understand why one cannot find an example using surfaces, odd-dimensional manifolds or any of $S^n$,$T^n$,$\mathbb{R}P^n$,$\mathbb{C}P^n$. I also understand why manifolds with $H^1(M;\mathbb{Z}_2)=0$ will not work, and why such a manifold (if it exists) cannot be null-cobordant.

My intuition on 4-manifolds (or god forbid anything higher dimensional) is not good enough to know where to look. Any insight as to where one should look for such a creature (or why the hell it should not exist) will be greatly appreciated.