  The paper investigates the coexistence and interplay of spin and mass
superfluidity in the antiferromagnetic spin-1 BEC. The hydrodynamical theory
describes the spin degree of freedom by the equations similar to the
Landau--Lifshitz--Gilbert theory for bipartite antiferromagnetic insulator. The
variables in the spin space are two subspins with absolute value $\hbar/2$,
which play the role of two sublattice spins in the antiferromagnetic
insulators. As well as in bipartite antiferromagnetic insulators, in the
antiferromagnetic spin-1 BEC there are two spin-wave modes, one is a gapless
Goldstone mode, another is gapped. The Landau criterion shows that in limit of
small total spin (two subspins are nearly antiparallel) instability of
supercurrents starts from the gapped mode. In the opposite limit of large total
spin (two subspins are nearly parallel) the gapless modes become unstable
earlier than the gapped one. Mass and spin supercurrents decay via phase slips,
when vortices cross streamlines of supercurrent. The vortices participating in
phase slips are nonsingular bicirculation vortices. They are characterized by
two topological charges, which are winding numbers describing circulations of
two angles around the vortex axis. The winding numbers can be half-integer. A
particular example of a half-integer vortex is a half-quantum vortex with the
superfluid velocity circulation $h/2m$. But the superfluid velocity circulation
is not a topological charge, and in general the quantum of this circulation can
be continuously tuned from 0 to $h/2m$.
