  We have studied the superconducting properties of the isostructural 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 indicate isotropic s-wave
superconductivity in TaReSi and multi-gap superconductivity in TaRuSi.
Zero-field $\mu$SR measurements, highly sensitive to very small magnetic
fields, and no evidence for spontaneous fields in the superconducting state of
TaReSi, whereas we observe small spontaneous fields that onset with
superconductivity indicating broken time-reversal symmetry (TRS)
superconductivity in TaRuSi. Using density functional theory calculations, we
find that spin-orbit coupling is relatively weak in TaRuSi and strong in
TaReSi. Using symmetry analysis, we attribute the broken time-reversal symmetry
(TRS) in TaRuSi to a non-unitary triplet pairing state. Such a state is not
allowed in the presence of strong spin-orbit coupling: our finding of no
evidence for broken TRS in TaReSi is consistent with this expectation.
