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We consider the scattering of particles (kinetic energy ε) by an obstacle which tunnels coherently between two positions (tunnel splitting Δ), for arbitrary values of $\varepsilon/\Delta$ and scattering strength U. The obstacle mimics two classical scatterers at fixed positions when $\varepsilon\lesssim \Delta$. Interference disappears when $\varepsilon \gg \Delta$, but can be recovered if the elastic-scattering channel is detected only. At intermediate values of ε there is a systematic interplay of the coherent tunnel motion and the ballistic particle motion. We show that the transmission probability can remain finite even in the limit $U\rightarrow\infty$ because the particle can evade the obstacle systematically. We discuss the realization of a quantum obstacle in mesoscopic systems.