  We describe the emergence of color superfluidity in ultra-cold fermions
induced by color-orbit and color-flip fields that transform a conventional
singlet-pairing s-wave system into an unconventional non-s-wave superfluid with
quintuplet pairing. We show that the tuning of interactions, color-orbit and
color-flip fields transforms a momentum-independent scalar order parameter into
an explicitly momentum-dependent tensor order parameter. We classify all
unconventional superfluid phases in terms of the {\it loci} of zeros of their
quasi-particle excitation spectrum in momentum space and we identify several
Lifshitz-type topological transitions. Furthermore, when boundaries between
phases are crossed, non-analyticities in the compressibility arise. We find a
quintuple point, which is also pentacritical, where four gapless superfluid
phases converge into a fully gapped superfluid phase.
