  Despite significant efforts over the last decade, the origin of the cosmic
ray positron excess has still not been unambiguously established. A popular
class of candidates are pulsars or pulsar wind nebulae but these cannot 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 the recent precision measurements by
AMS-02 of cosmic ray protons, helium, positrons and antiprotons, as well as of
various primary and secondary nuclei. The only notable deviation is an excess
of antiprotons around ~10 GeV. This model thus provides an economical
explanation of the spectra of all secondary species -- from a single
well-motivated population of sources.
