Around 400 Myr after the big bang, the ultraviolet emission from star-forming galaxies reionized the Universe. Ionizing radiation (Lyman continuum, LyC) is absorbed by cold neutral hydrogen gas (H i) within galaxies, hindering the escape of LyC photons. Since the H i reservoir of LyC emitters has never been mapped, major uncertainties remain on how LyC photons escape galaxies and ionize the intergalactic medium. We have directly imaged the neutral gas in the nearby reionization-era analogue galaxy Haro 11 with the 21 cm line to identify the mechanism enabling ionizing radiation escape. We find that merger-driven interactions have caused a bulk offset of the neutral gas by about 6 kpc from the centre of the galaxy, where LyC emission production sites are located. This could facilitate the escape of ionizing radiation into our line of sight. Galaxy interactions can cause both elevated LyC production and large-scale displacement of H i from the regions where these photons are produced. They could contribute to the anisotropic escape of LyC radiation from galaxies and the reionization of the Universe. We argue for a systematic assessment of the effect of environment on LyC production and escape.