The complex structure of the Remeika phases, the intriguing quantum states they display, and their low carrier concentrations are a strong motivation to study the nature of their superconducting phases. In this work, the microscopic properties of the superconducting phase of single-crystalline Lu3Os4Ge13 are investigated by muon-spin relaxation and rotation (µSR) measurements. The zerofield µSR data reveal the presence of spontaneous static or quasi-static magnetic fields in the superconducting state, breaking time-reversal symmetry; the associated internal magnetic field scale is found to be exceptionally large ( 0.18 mT). Furthermore, transverse-field µSR measurements in the vortex state of Lu3Os4Ge13 imply a complex gap function with significantly different strengths on different parts of the Fermi surface. While our measurements do not completely determine the order parameter, they strongly indicate that electron-electron interactions are essential to stabilizing pairing in the system, thus, demonstrating its unconventional nature.