In a multiwavelength programme dedicated to identifying optical counterparts of faint persistent X-ray sources in the Galactic bulge, we find an accurate X-ray position of SAX J1712.6-3739 through Chandra observations, and discover its faint optical counterpart using our data from EFOSC2 on the ESO 3.6-m telescope. We find this source to be a highly extincted neutron star low-mass X-ray binary (LMXB) with blue optical colours. We serendipitously discover a relatively bright and large bow shock shaped nebula in our deep narrow-band Hα imaging, most likely associated with the X-ray binary. A nebula like this has never been observed before in association with a LMXB, and as such provides a unique laboratory to study the energetics of accretion and jets. We put forward different models to explain the possible ways the LMXB may form this nebulosity, and outline how they can be confirmed observationally.