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Vela X-1 is a runaway X-ray binary system hosting a massive donor star, whose strong stellar wind creates a bow shock as it interacts with the interstellar medium. This bow shock has previously been detected in H$\alpha$ and IR, but, similar to all but one bow shock from a massive runaway star (BD+43$^{\rm o}$3654), has escaped detection in other wavebands. We report on the discovery of $1.3$ GHz radio emission from the Vela X-1 bow shock with the MeerKAT telescope. The MeerKAT observations reveal how the radio emission closely traces the H$\alpha$ line emission, both in the bow shock and in the larger-scale diffuse structures known from existing H$\alpha$ surveys. The Vela X-1 bow shock is the first stellar-wind-driven radio bow shock detected around an X-ray binary. In the absence of a radio spectral index measurement, we explore other avenues to constrain the radio emission mechanism. We find that thermal/free-free emission can account for the radio and H$\alpha$ properties, for a combination of electron temperature and density consistent with earlier estimates of ISM density and the shock enhancement. In this explanation, the presence of a local ISM over-density is essential for the detection of radio emission. Alternatively, we consider a non-thermal/synchrotron scenario, evaluating the magnetic field and broad-band spectrum of the shock. However, we find that exceptionally high fractions ($\gtrsim 13$%) of the kinetic wind power would need to be injected into the relativistic electron population to explain the radio emission. Assuming lower fractions implies a hybrid scenario, dominated by free-free radio emission. Finally, we speculate about the detectability of radio bow shocks and whether it requires exceptional ISM or stellar wind properties., Comment: 17 pages including appendices, 12 figures; accepted for publication in MNRAS. Updated version with minor corrections to reference list