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 belts mass or the binarys 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 systems 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
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