  We aim to see whether debris belts evolving in between two stars may be
impacted by the presence of the companion and whether this leaves any
detectable signature that could be observed with current or future instruments.
We consider a circumprimary parent body (PB) planetesimal belt that is placed
just inside the stability limit between the 2 stars and use the DyCoSS code to
follow the coupled dynamical and collisional evolution of the dust produced by
this PB belt. We explore several free parameters such as the belt's mass or the
binary's mass ratio and orbit. We use the GraTeR package to produce 2-D
luminosity maps and system-integrated SEDs. We confirm a preliminary result
obtained by earlier DyCoSS studies, which is that the coupled effect of
collisional activity, binary perturbations and stellar radiation pressure
maintains a halo of small grains in the dynamically unstable region between the
2 stars. In addition, several spatial structures are identified, notably a
single spiral arm stretching all the way from the PB belt to the companion
star. We also identify a fainter and more compact disc around the secondary
star, which is non-native and feeds off small grains from the unstable halo.
Both the halo, spiral arm and secondary disc should be detectable on resolved
images by instruments with capacities similar to SPHERE. The system as a whole
is depleted in small grains when compared to a companion-free case. This
depletion leaves an imprint on the system's integrated SED, which appears
colder than for the same parent body belt around a single star. This new
finding could explain why the SED-derived location $r_{disc}$ of some
unresolved discs-in-binaries places their primary belt in the dynamically
"forbidden" region between the 2 stars: this apparent paradox could indeed be
due to overestimating $r_{disc}$ when using empirical prescriptions valid for a
single star case
