Semi-regular (SR) variables are not a homogeneous class and their variability is often explained due to pulsations and/or binarity. This study focuses on IRAS 19135+3937, an SRd variable with an infrared excess indicative of a dusty disc. A time series of high-resolution spectra, UBV photometry as well as a very accurate light curve obtained by the Kepler satellite, allowed us to study the object in unprecedented detail. We discovered it to be a binary with a period of 127 d. The primary has a low surface gravity and an atmosphere depleted in refractory elements. This combination of properties unambiguously places IRAS 19135+3937 in the subclass of post-asymptotic giant branch stars with dusty discs. We show that the light variations in this object cannot be due to pulsations, but are likely caused by the obscuration of the primary by the circumbinary disc during orbital motion. Furthermore, we argue that the double-peaked Fe emission lines provide evidence for the existence of a gaseous circumbinary Keplerian disc inside the dusty disc. A secondary set of absorption lines has been detected near light minimum, which we attribute to the reflected spectrum of the primary on the disc wall, which segregates due to the different Doppler shift. This corroborates the recent finding that reflection in the optical by this type of discs is very efficient. The system also shows a variable H α profile indicating a collimated outflow originating around the companion. IRAS 19135+3937 thus encompasses all the major emergent trends about evolved disc systems, that will eventually help to place these objects in the evolutionary context.