  We study the superconducting state of multi-orbital spin-orbit coupled
systems in the presence of an orbitally driven inversion asymmetry assuming
that the inter-orbital attraction is the dominant pairing channel. Although the
inversion symmetry is absent, we show that superconducting states that avoid
mixing of spin-triplet and spin-singlet configurations are allowed, and
remarkably, spin-triplet states that are topologically nontrivial can be
stabilized in a large portion of the phase diagram. The orbital-dependent
spin-triplet pairing generally leads to topological superconductivity with
point nodes that are protected by a nonvanishing winding number. We demonstrate
that the disclosed topological phase can exhibit Lifshitz-type transitions upon
different driving mechanisms and interactions, e.g., by tuning the strength of
the atomic spin-orbit and inversion asymmetry couplings or by varying the
doping and the amplitude of order parameter. Such distinctive signatures of the
nodal phase manifest through an extraordinary reconstruction of the low-energy
excitation spectra both in the bulk and at the edge of the superconductor.
