The recent discovery of the binary containing the slow pulsar PSR 1718-19 orbiting around a low-mass companion star adds new light on the characteristics of binary pulsars. The properties of the radio eclipses of PSR 1718-19 are the most striking observational characteristics of this system. The surface of the companion star produces a mass outflow which leaves only a small 'window' in orbital phase for the detection of PSR 1718-19 around 400 MHz. At this observing frequency, PSR 1718-19 is clearly observable only for about 1 hr out of the total 6.2 hr orbital period. The aim of this Letter is twofold: (1) to model the hydrodynamical behavior of the eclipsing material from the companion star of PSR 1718-19 and (2) to argue that a population of binary slow pulsars might have escaped detection in pulsar surveys carried out at 400 MHz. The possible existence of a population of partially or totally hidden slow pulsars in binaries will have a strong impact on current theories of binary evolution of neutron stars.