This is probably overkill, but I couldn't resist advertising a preprint that Diarmuid Crowley and I recently posted to the arXiv: https://arxiv.org/abs/1802.01296
In the final section we discuss examples due to Teichner of closed $6$-manifolds with extraordinary cohomological properties. These examples are constructed as sphere bundles of rank $3$ vector bundles over closed $4$-manifolds. In particular, in Proposition 5.9 we consider the total space of a sphere bundle $S^2\to N\to M$ where the base manifold has $$ \pi_1(M)=\mathbb{Z}/8\rtimes\mathbb{Z}/2. $$ The manifold $N$ is not spin$^c$. This follows from the main theorem of our paper, since $N$ has a cohomology class $x\in H^2(N;\mathbb{Z}/2)$ such that $\beta(x^2)$ is not a multiple of $\beta(x)$ (here $\beta$ denotes the Bockstein from mod $2$ to integral cohomology).
Now, if we pull this sphere bundle back to the universal cover $\widetilde{M}\to M$, the total space $\widetilde{N}$ will be a finite cover of $N$, and will be the sphere bundle of the vector bundle $\pi:\widetilde{E}\to \widetilde{M}$ which is the pullback of the bundle $E\to M$ defining $N$. It remains to show that $\widetilde{N}$ is spin$^c$.
A general argument about tangent bundles of sphere bundles gives a bundle isomorphism $$ T\widetilde{N}\oplus\mathbb{R}\cong \pi^*T\widetilde{M} \oplus \pi^*\widetilde{E}, $$ which (since $\widetilde{M}$ and $\widetilde{E}$ are orientable) gives $$w_2(\widetilde{N})= \pi^*w_2(\widetilde{M}) + \pi^*w_2(\widetilde{E}).$$ Therefore, $$\beta w_2(\widetilde{N})= \pi^*\beta w_2(\widetilde{M}) + \beta \pi^*w_2(\widetilde{E}).$$ The first term on the right vanishes since an orientable $4$-manifold is spin$^c$. The second term vanishes since $\widetilde{E}$ is an orientable bundle of rank $3$, and therefore $\beta w_2(\widetilde{E})=e(\widetilde{E})$ equals the Euler class, which pulls back to zero in the sphere bundle. Hence $\beta w_2(\widetilde{N})=0$, and $\widetilde{N}$ is spin$^c$ as claimed.