We have studied the superconducting properties of the ternary
noncentrosymmetric superconductors TaXSi (X = Re, Ru), with the help of muon
spin rotation/relaxation ({\mu}SR) and density functional theory calculations.
Our transverse-field {\mu}SR measurements reveal isotropic s-wave
superconductivity in TaReSi and multi-gap superconductivity with gap nodes in
TaRuSi. Zero-field {\mu}SR measurements, highly sensitive to very small
magnetic fields find no evidence for spontaneous fields in the superconducting
state of TaReSi, whereas we observe spontaneous fields which onset with
superconductivity indicating broken time reversal symmetry (TRS)
superconductivity in TaRuSi. Broken TRS in weakly coupled TaRuSi can be
attributed to a non-unitary triplet pairing state, while in TaReSi, this state
is suppressed due to strong anti-symmetric spin orbital coupling. Our results
in TaXSi demonstrate that the strength of spin orbit coupling can be
responsible for stabilizing unconventional superconductivity.Comment: 10 Pages, 5 Figure