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When a superconductor is cooled below its critical temperature Tc the phase of the electronic wave function gets locked to a particular value. This leads to quantum effects on the scale of everyday objects which underpin superconducting quantum technologies. In some unconventional superconductors this form of symmetry breaking is concomitant with additional breaking of time-reversal symmetry (TRS) indicating that the superconducting state is intrinsically magnetic. Such systems are expected to have important applications in spintronics and topological quantum computing, however this is hindered by the lack of a general theory of unconventional superconductivity which is normally associated with strong electron correlations or fluctuations of competing ordered phases. Recently, however, TRS breaking has been reported in seemingly ordinary superconductors where such exotic physics are not at play, including the chemical element rhenium. Here we show that TRS breaking in Re is due to a form of mixed singlet-triplet pairing that has an atomic-scale magnetic texture. Rather than assuming an unconventional pairing interaction from the outset, we couple a conventional pairing model with an ab initio description of the system's magnetism and relativistic electronic structure. We find that a triplet pairing component emerges spontaneously due to spin-orbit coupling, without further symmetry breaking. When an additional pairing term operating in this channel is added in order to make our theory self-consistent a phase with broken time-reversal symmetry emerges. Through computer experiments we identify the non-symmorphic crystal structure as the key ingredient of this exotic new state. Our approach represents a significant departure from previous attempts at understanding symmetry-breaking in unconventional superconductors, yet it describes experimental data quantitatively with only two adjustable parameters, showing that unconventional superconductivity can be more ubiquitous than hitherto assumed.