  In 5$d^2$ Mott insulators with strong spin-orbit coupling, the lowest
pseudospin states form a non-Kramers doublet, which carries quadrupolar and
octupolar moments. A family of double perovskites where magnetic ions form a
face-centered cubic (fcc) lattice was suggested to unveil an octupolar order
offering a rare example in $d$-orbital systems. The proposed order requires a
ferromagnetic (FM) octupolar interaction, since the antiferromagnetic (AFM)
Ising model is highly frustrated on the fcc lattice. A microscopic model was
recently derived for various lattices: for an edge-sharing octahedra geometry,
AFM Ising octupolar and bond-dependent quadrupolar interactions were found when
only dominant inter- and intraorbital hopping integrals were taken into
account. Here we investigate all possible intra- and interorbital exchange
processes and report that interference of two intraorbital exchanges generates
a FM octupolar interaction. Applying the strong-coupling expansion results
together with tight-binding parameters obtained by density functional theory,
we estimate the exchange interactions for the osmium double perovskites,
Ba$_2$BOsO$_6$ (B=Mg,Cd, and Ca). Using classical Monte Carlo simulations, we
find that these systems are close to the phase boundary between AFM type-I
quadrupole and FM octupole orders. We also find that exchange processes beyond
second-order perturbation theory including virtual processes via
pseudospin-triplet states may stabilize an octupolar order.
