  We investigate a three component fermion mixture in the presence of weak
attractive interactions. We use a combination of the equation of motion and the
Gaussian variational mean-field approaches, which both allow for simultaneous
superfluid and magnetic ordering in an unbiased way, and capture the interplay
between the two order parameters. This interplay significantly modifies the
phase diagram, especially the superfluid-normal phase boundaries. In the close
vicinity of the critical temperature and for small chemical potential
imbalances, strong particle-hole symmetry breaking leads to a phase diagram
similar to the one predicted by Cherng et al. [Phys. Rev. Lett. 99, 130406
(2007)], however, the overall phase diagram is markedly different: new chemical
potential-driven first and second order transitions and triple points emerge as
well as more exotic second order multicritical points, and bicritical lines
with O(2,2) symmetry. We identify the terms which are necessary to capture this
complex phase diagram in a Ginzburg-Landau approach, and determine the
corresponding coefficients.
