  Observations of soft gamma rays emanating from the Milky Way from
SPI/\textit{INTEGRAL} reveal the annihilation of $\sim2\times10^{43}$ positrons
every second in the Galactic bulge. The origin of these positrons, which
annihilate to produce a prominent emission line centered at 511 keV, has
remained mysterious since their discovery almost 50 years ago. A plausible
origin for the positrons is in association with the intense star formation
ongoing in the Galactic center. Moreover, there is strong evidence for a
nuclear outflow in the Milky Way. We find that advective transport and
subsequent annihilation of positrons in such an outflow cannot simultaneously
replicate the observed morphology of positron annihilation in the Galactic
bulge and satisfy the requirement that $90$ per cent of positrons annihilate
once the outflow has cooled to $10^4\,\mathrm{K}$.
