  Bose-Einstein condensates (BECs) of positronium (Ps) have been of
experimental and theoretical interest due to their potential application as the
gain medium of a $\gamma$-ray laser. Ps BECs are intrinsically spinor due to
the presence of ortho-positronium (o-Ps) and para-positronium (p-Ps), whose
annihilation lifetimes differ by three orders of magnitude. In this paper, we
study the spinor dynamics and annihilation processes in the p-Ps/o-Ps system
using both solutions of the time-dependent Gross-Pitaevskii equations and a
semiclassical rate-equation approach. The spinor interactions have an $O(4)$
symmetry which is broken to $SO(3)$ by an internal energy difference between
o-Ps and p-Ps. For an initially unpolarized condensate, there is a threshold
density of $\approx 10^{19}$ cm$^{-3}$ at which spin mixing between o-Ps and
p-Ps occurs. Beyond this threshold, there are unstable spatial modes
accompanied by spin mixing. To ensure a high production yield above the
critical density, a careful choice of external field must be made to avoid the
spin mixing instability.
