  Recent $\mu$SR measurements revealed remarkable signatures of spontaneous
magnetism coexisting with superconductivity in elemental rhenium. Here we
provide a quantitative theory that uncovers the nature of the superconducting
instability by incorporating every details of the electronic structure together
with spin-orbit coupling and multi-orbital physics. We show that conventional
$s$-wave superconductivity combined with strong spin-orbit coupling is inducing
even-parity odd-orbital spin triplet Cooper pairs, and in presence of a screw
axis Cooper pairs' migration between the induced equal-spin triplet component
leads to an exotic magnetic state.
