  The pairing symmetry of interacting Dirac fermions on the $\pi$-flux lattice
is studied with the determinant quantum Monte Carlo and numerical linked
cluster expansion methods. The extended $s^*$- (i.e. extended $s$-) and d-wave
pairing symmetries, which are distinct in the conventional square lattice, are
degenerate under the Landau gauge. We demonstrate that the dominant pairing
channel at strong interactions is an exotic $ds^*$-wave phase consisting of
alternating stripes of $s^*$- and d-wave phases. A complementary mean-field
analysis shows that while the $s^*$- and d-wave symmetries individually have
nodes in the energy spectrum, the $ds^*$ channel is fully gapped. The results
represent a new realization of pairing in Dirac systems, connected to the
problem of chiral d-wave pairing on the honeycomb lattice, which might be more
readily accessed by cold-atom experiments.
