  Despite significant efforts over the past decade, the origin of the cosmic
ray positron excess has still not been unambiguously established. A popular
class of candidate sources are pulsars or pulsar wind nebulae but these cannot
also account for the observed hard spectrum of cosmic ray antiprotons. We
revisit the alternative possibility that the observed high-energy positrons are
secondaries created by spallation in supernova remnants during the diffusive
shock acceleration of the primary cosmic rays, which are further accelerated by
the same shocks. The resulting source spectrum of positrons at high energies is
then naturally harder than that of the primaries, as is the spectrum of other
secondaries such as antiprotons. We present the first comprehensive
investigation of the full parameter space of this model -- both the source
parameters as well as those governing galactic transport. Various
parameterisations of the cross sections for the production of positrons and
antiprotons are considered, and the uncertainty in the model parameters
discussed. We obtain an excellent fit to recent precision measurements by
AMS-02 of cosmic ray protons, helium, positrons and antiprotons, as well as of
various primary and secondary nuclei. This model thus provides an economical
explanation of the spectra of all secondary species -- from a single
well-motivated population of sources.
