  The Feynman amplitude for the decay of the $J/\psi$ meson into
baryon-antibaryon can be written as a sum of three sub-amplitudes: a purely
strong, a purely electromagnetic and a mixed strong-electromagnetic. Assuming
that the strong and mixed strong-electromagnetic sub-amplitudes have the same
phase, the branching ratio of the decay contains an interference term that
depends on the relative phase $\varphi$ between strong and electromagnetic
sub-amplitudes. In this work we calculate this phase, by using an effective
strong Lagrangian density and considering, as final states, pairs of baryons,
$\mathcal{B}\overline{\mathcal{B}}$, belonging to the spin-1/2 SU(3) octet.
Moreover, we obtain the purely strong, purely electromagnetic and mixed
strong-electromagnetic contributions to the total branching ratio and hence the
moduli of the corresponding sub-amplitudes, for each pair of baryons. Of
particular interest is the mixed strong-electromagnetic contribution that, not
only is determined for the first time, but it is proven to be crucial, in the
framework of our model, for the correct description of the decay mechanism.
Finally we use the purely electromagnetic branching ratio to calculate the Born
non-resonant cross section of the annihilation processes $e^+ e^- \to
\mathcal{B}\overline{\mathcal{B}}$ at the $J/\psi$ mass. By taking advantage
from all available data, we obtain the relative phase between strong and
electromagnetic sub-amplitudes: $\varphi = (73\pm 8)^\circ$.
