  Observations of binary pulsars and pulsars in globular clusters suggest that
at least some pulsars must receive weak natal kicks at birth. If all pulsars
received strong natal kicks above \unit[50]{\kms}, those born in globular
clusters would predominantly escape, while wide binaries would be disrupted. On
the other hand, observations of transverse velocities of isolated radio pulsars
indicate that only $5\pm2\%$ have velocities below \unit[50]{\kms}. We explore
this apparent tension with rapid binary population synthesis modelling. We
propose a model in which supernovae with characteristically low natal kicks
(e.g., electron-capture supernovae) only occur if the progenitor star has been
stripped via binary interaction with a companion. We show that this model
naturally reproduces the observed pulsar speed distribution and without
reducing the predicted merging double neutron star yield. We estimate that the
zero-age main sequence mass range for non-interacting progenitors of
electron-capture supernovae should be no wider than ${\approx}0.2 M_\odot$.
