The binaries PSR J1141-6545 and PSR B2303+46 each appear to contain a white dwarf which formed before the neutron star. We describe an evolutionary pathway to produce these two systems. In this scenario, the primary transfers its envelope onto the secondary which is then the more massive of the two stars, and indeed sufficiently massive later to produce a neutron star via a supernova. The core of the primary produces a massive white dwarf which enters into a common envelope with the core of the secondary when the latter evolves off the main sequence. During the common envelope phase, the white dwarf and the core of the secondary spiral together as the envelope is ejected. The evolutionary history of PSR J1141-6545 and PSR B2303+46 differ after this phase. In the case of PSR J1141--6545, the secondary (now a helium star) evolves into contact transferring its envelope onto the white dwarf. We propose that the vast majority of this material is in fact ejected from the system. The remains of the secondary then explode as a supernova producing a neutron star. Generally the white dwarf and neutron star will remain bound in tight, often eccentric, systems resembling PSR J1141-6545. These systems will spiral in and merge on a relatively short timescale and may make a significant contribution to the population of gamma ray burst progenitors. In PSR B2303+46, the helium-star secondary and white dwarf never come into contact. Rather the helium star loses its envelope via a wind, which increases the binary separation slightly. Only a small fraction of such systems will remain bound when the neutron star is formed (as the systems are wider). Those systems which are broken up will produce a population of high-velocity white dwarfs and neutron stars.
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