  The progenitor systems and explosion mechanism of Type Ia supernovae are
still unknown. Currently favoured progenitors include double-degenerate systems
consisting of two carbon-oxygen white dwarfs with thin helium shells. In the
double-detonation scenario, violent accretion leads to a helium detonation on
the more massive primary white dwarf that turns into a carbon detonation in its
core and explodes it. We investigate the fate of the secondary white dwarf,
focusing on changes of the ejecta and observables of the explosion if the
secondary explodes as well rather than survives. We simulate a binary system of
a $1.05\,\mathrm{M_\odot}$ and a $0.7\,\mathrm{M_\odot}$ carbon-oxygen white
dwarf with $0.03\,\mathrm{M_\odot}$ helium shells each. We follow the system
self-consistently from inspiral to ignition, through the explosion, to
synthetic observables. We confirm that the primary white dwarf explodes
self-consistently. The helium detonation around the secondary white dwarf,
however, fails to ignite a carbon detonation. We restart the simulation
igniting the carbon detonation in the secondary white dwarf by hand and compare
the ejecta and observables of both explosions. We find that the outer ejecta at
$v~>~15\,000$\,km\,s$^{-1}$ are indistinguishable. Light curves and spectra are
very similar until $\sim~40\,\mathrm{d}$ after explosion and the ejecta are
much more spherical than violent merger models. The inner ejecta differ
significantly slowing down the decline rate of the bolometric light curve after
maximum of the model with a secondary explosion by $\sim20$ per cent. We expect
future synthetic 3D nebular spectra to confirm or rule out either model.
