  We study the rheology of a soft particulate system where the inter-particle
interactions are weakly attractive. Using extensive molecular dynamics
simulations, we scan across a wide range of packing fractions ($\phi$),
attraction strengths ($u$) and imposed shear-rates ($\dot{\gamma}$). In
striking contrast to repulsive systems, we find that at small shear-rates
generically a fragile isostatic solid is formed even if we go to $\phi \ll
\phi_J$. Further, with increasing shear-rates, even at these low $\phi$,
non-monotonic flow curves occur which lead to the formation of persistent
shear-bands in large enough systems. By tuning the damping parameter, we also
show that inertia plays an important role in this process. Furthermore, we
observe enhanced particle dynamics in the attraction-dominated regime as well
as a pronounced anisotropy of velocity and diffusion constant, which we take as
precursors to the formation of shear bands. At low enough $\phi$, we also
observe structural changes via the interplay of low shear-rates and attraction
with the formation of micro-clusters and voids. Finally, we characterize the
properties of the emergent shear bands and thereby, we find surprisingly small
mobility of these bands, leading to prohibitely long time-scales and extensive
history effects in ramping experiments.
