In the past decade, the large area telescope (LAT) aboard Fermi has uncovered more than three thousand gamma-ray sources. As a consequence, compact binary millisecond pulsars (with orbital periods shorter than about a day) are becoming a sizeable and rapidly-growing pulsar population. Pulsars are the main Galactic class among the known gamma-ray sources, yet nearly a thousand LAT sources remain unidentified. We present the first results from our COBIPULSE survey: a systematic search for optical variable and X-ray counterparts to Fermi-LAT pulsar candidates. These include the discovery of a new redback candidate in a 21-hr orbit (3FGL J0212.1+5320), the brightest of its kind in the optical band. We will also present the recent finding of a very massive neutron star in the millisecond pulsar PSR J2215+5135, with a record mass of 2.3 Solar masses.