Any experimental evidence of nucleons paired in spin-triplet states will confirm the existence of an exotic phase of nuclear systems. This type of nuclear superfluidity has been hypothesized in heavy nuclei, where the antagonizing spin-orbit effects are damped, and there it oftentimes coexists with traditional spin-singlet pairing, leading to the possibility of mixed-spin pairing. Realistic nuclear deformation, not considered in such studies, could make-or-break these proposals, since its effect on triplet pairing, and the competition (and coexistence) of the two superfluid phases, was expected to be crucial. We report on a thorough study on the effect of deformation on triplet, singlet, and mixed-spin pairing in the relevant region of the nuclear chart. We find that, at low isospin asymmetries, spin-triplet pairing is enhanced by deformation, while below the proton-drip line, the novel superfluid phase survives alongside the usual spin-singlet pairing. These results suggest that spin-triplet superfluidity exists in realistic nuclei and can be probed in the lab.