The rise in the positron fraction, observed by PAMELA, Fermi-LAT and most
recently by AMS-02, has created a lot of interest, fuelled by speculations about an origin in dark matter annihilation in the Galactic halo. However, other channels, e.g. antiprotons or gamma-rays, now severely constrain dark matter interpretations, thus requiring astrophysical sources of positrons. We have investigated the possibility that supernova remnants, the most likely sources of Galactic cosmic rays, can in fact also produce a hard spectrum of secondary positrons, by spallation and acceleration at the shock. This mechanism is guaranteed if hadronic CRs are present and would also lead to observable signatures in other secondary channels like the boron-to-carbon or antiproton-to-proton ratios. If such features were borne out by upcoming AMS-02 data, this would rule out other explanations.