Standard binary evolution models of pulsars in very circular orbits can explain systems with lighter companions and weaker magnetic fields: they are descended from long-lived Low Mass X-ray Binaries with Roche-lobe overflow. By contrast, those having more massive companions and larger magnetic fields are descendants of Intermediate Mass systems, that experienced a common envelope phase. PSR J1744-3922 is a Galactic field binary pulsar exhibiting an unusual combination of characteristics compared to other binary pulsars with low-mass members in circular orbits: it has a relatively long spin period (172 ms), a relatively high magnetic field (1.7×1010 G), a short orbital period (4.6 h) and a light companion (Mmin = 0.08Msolar). These properties are not self-consistently explained by standard evolution models and so suggest that PSR J1744-3922 has a peculiar evolutionary history. We identify some other previously catalogued Galactic field binary pulsars with similar properties and propose they are potential members of a new class of binary pulsars. Such pulsars could possibly be formed by the recycling of a pulsar originally having a magnetar-strength magnetic field, accretion-induced collapse of a massive white dwarf or a special case of common envelope accretion from a low-mass star.