  Recent experiments report that slowly-sheared noncolloidal particle
suspensions can exhibit unexpected rate($\omega$)-dependent complex viscosities
in oscillatory shear, despite a constant relative viscosity in steady shear.
Using a minimal hydrodynamic model, we show that a weak interparticle
attraction reproduces this behavior. At volume fractions $\phi=20\sim50$%, the
complex viscosities in both experiments and simulations display power-law
reductions in shear, with a $\phi$-dependent exponent maximum at $\phi=40$%,
resulting from the interplay between hydrodynamic, collision and adhesive
interactions. Furthermore, this rate dependence is accompanied by diverging
particle diffusivities and pronounced cluster formations even at small
oscillation amplitudes $\gamma_0$. Previous studies established that
suspensions transition from reversible absorbing states to irreversible
diffusing states when $\gamma_0$ exceeds a $\phi$-dependent critical value
$\gamma_{0,\phi}^c$. Here, we show that a second transition to irreversibility
occurs below an $\omega$-dependent critical amplitude, $\gamma_{0,\omega}^c
\leq \gamma_{0,\phi}^c$, in the presence of weak attractions.
