  The noncentrosymmetric superconductor Re$_{24}$Ti$_{5}$, a time-reversal
symmetry (TRS) breaking candidate with $T_c = 6$\,K, was studied by means of
muon-spin rotation/relaxation ($\mu$SR) and tunnel-diode oscillator (TDO)
techniques. At a macroscopic level, its bulk superconductivity was investigated
via electrical resistivity, magnetic susceptibility, and heat capacity
measurements. The low-temperature penetration depth, superfluid density and
electronic heat capacity all evidence an $s$-wave coupling with an enhanced
superconducting gap. The spontaneous magnetic fields revealed by zero-field
$\mu$SR below $T_c$ indicate a time-reversal symmetry breaking and thus the
unconventional nature of superconductivity in Re$_{24}$Ti$_{5}$. The
concomitant occurrence of TRS breaking also in the isostructural Re$_6$(Zr,Hf)
compounds, hints at its common origin in this superconducting family and that
an enhanced spin-orbital coupling does not affect pairing symmetry.
