We demonstrate time-domain observation of the core-hole decay of xenon atoms based on photoelectron interferometry, using phase-coherent extreme ultraviolet pulse pairs generated from a synchrotron light source. By utilizing a tandem undulator configuration, we induced interference between inner-shell photoelectron wave packets and measured the resulting Ramsey fringes via photoelectron spectroscopy. The contrast of these fringes exhibits a clear decay as a function of the femtosecond-scale time delay between the two pulses, directly reflecting the temporal evolution of the core-hole. This synchrotron-based interferometry holds the potential to probe the ultrafast electron dynamics with element- and site-selectivity.