  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
Lu$_3$Os$_4$Ge$_{13}$ are investigated by muon-spin relaxation and rotation
($\mu$SR) measurements. The zero-field $\mu$SR 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 ($\approx$ 0.18~mT). Furthermore,
transverse-field $\mu$SR measurements in the vortex state of
Lu$_3$Os$_4$Ge$_{13}$ 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.
