X-ray free-electron lasers (FELs) deliver ultrabright X-ray pulses, but not the sequences of phase-coherent pulses required for time-domain interferometry and control of quantum states. For conventional split-and-delay schemes to produce such sequences, the challenge stems from extreme stability requirements when splitting angstrom ngstrom wavelength beams, where the tiniest pathlength differences introduce phase jitter. We describe an FEL mode based on selective electron-bunch degradation and transverse beam shaping in the accelerator, combined with a self-seeded photon emission scheme. Instead of splitting the photon pulses after their generation by the FEL, we split the electron bunch in the accelerator, prior to photon generation, to obtain phase-locked X-ray pulses with subfemtosecond duration. Time-domain interferometry becomes possible, enabling the concomitant program of classical and quantum optics experiments with X-rays. The scheme leads to scientific benefits of cutting-edge FELs with attosecond and/or high-repetition rate capabilities, ranging from the X-ray analog of Fourier transform infrared spectroscopy to damage-free measurements.