We report the observation of an intriguing behavior in the transport properties of nanodevices operating in a regime between the Fabry-Pérot and the Kondo limits. Using ultrahigh quality nanotube devices, we study how the conductance oscillates when sweeping the gate voltage. Surprisingly, we observe a fourfold enhancement of the oscillation period upon decreasing temperature, signaling a crossover from single-electron tunneling to Fabry-Pérot interference. These results suggest that the Fabry-Pérot interference occurs in a regime where electrons are correlated. The link between the measured correlated Fabry-Pérot oscillations and the SU(4) Kondo effect is discussed.