Context. Some RV Tauri stars show a long-term photometric variability in their mean magnitudes. DF Cygni (DF Cyg), the only RV Tauri star in the original Kepler field, and the prototype RV Tauri (RV Tau) are two such stars. Aims. The focus of this paper is on two famous but still poorly understood RV Tauri stars: RV Tau and DF Cyg. We aim to confirm their suspected binary nature and derive their orbital elements to investigate the impact of their orbits on the evolution of these systems. This research is embedded in a wider endeavour to study binary evolution of low- and intermediate-mass stars. Methods. The high amplitude pulsations were cleaned from the radial-velocity data to better constrain the orbital motion, allowing us to obtain accurate orbital parameters. We also analysed the photometric time series of both stars using a Lomb-Scargle periodogram. We used Gaia Data Release 2 (DR2) parallaxes in combination with the spectral energy distributions (SEDs) to compute their luminosities. These luminosities were complemented with the ones we computed using a period-luminosity-colour (PLC) relation for RV Tauri stars. The ratio of the circumstellar infrared (IR) flux to the photospheric flux obtained from the SEDs was used to estimate the orbital inclination of each system. Results. DF Cyg and RV Tau are binaries with spectroscopic orbital periods of 784 ± 16 days and 1198 ± 17 days, respectively. These orbital periods are found to be similar to the long-term periodic variability in the photometry, indicating that binarity indeed explains the long-term photometric variability. The SEDs of these systems indicate the presence of a circumbinary disc. Our line of sight grazes the dusty disc, which causes the photometric flux from the star to extinct periodically with the orbital period. Our derived orbital inclinations enabled us to obtain accurate companion masses for DF Cyg and RV Tau, and these were found to be 0.6 ± 0.1 M_⊙ and 0.7 ± 0.1 M_⊙, respectively. The derived luminosities suggest that RV Tau is a post asymptotic giant branch (post-AGB) binary, while DF Cyg is likely a post red giant branch (post-RGB) binary. Analysis of the Kepler photometry of DF Cyg revealed a power spectrum with side lobes around the fundamental pulsation frequency. This modulation corresponds to the spectroscopic orbital period and hence to the long-term photometric period. Finally we report on the evidence of high velocity absorption features related to the H_α profile in both objects, indicating outflows launched from around the companion.