It's known that all oriented 4-manifolds admit a $Spin^c$ structure, ie. a spin structure on $TX\oplus\mathcal{L}$ for some complex line bundle $\mathcal{L}$.

A usual generalization of this structure to unorientable manifolds is to ask for a spin structure on $TX\oplus\mathcal{L}\oplus\mathcal{E}$ where $\mathcal{L}$ is again a complex line and now $\mathcal{E}$ is a real line bundle (which the whole bundle being oriented will force to be the orientation line). This is called a $Pin^c$ structure. In cohomology, it amounts to an integral lift of $w_2$.

Unfortunately, not all 4-manifolds admit a $pin^c$ structure, eg. $\mathbb{RP}^2 \times \mathbb{RP}^2$. This is easy to see by a computation of $w_2$.

There is another generalization I'll call a $Pin^{\tilde c}$ structure. In this version, $\mathcal{L}$ and $\mathcal{E}$ combine into a real 2-plane bundle. In cohomology it amounts to a *twisted* integral lift of $w_2$.

So, do all 4-manifolds admit a twisted integral lift of $w_2$?

Here are some edits in response to Qiaochu's comment. An integral lift of $w_2$ is an element of $H^2(X,\mathbb{Z})$ mapping to $w_2$ in $H^2(X,\mathbb{Z}/2)$. A twisted integral lift lives instead in $H^2(X,\mathbb{Z}^{w_1})$, with local coefficients in the orientation line.