For pendular suspensions with particles in contact with immiscible secondary liquid bridges, the shear field significantly influences particle aggregates and networks. In this work, we study the structure of the pendular network and how the structure changes under large-amplitude-oscillatory shear. Using rheology and optical microscopy, we found unique network destruction followed by reconstruction with increasing strain. Two processes show different shear-field dependencies, strain-rate dependency for destruction and strain dependency for reconstruction. A nonequilibrium state diagram is constructed to show the phase behavior, where the critical particle concentration of sol-gel transition is dependent on the shear history and may depend on shear strain nonmonotonically. Two different mechanisms, shear-induced network breakdown at low strain and shear-induced agglomeration at high strain, are suggested to describe the nonmonotonic critical concentration under the upward strain sweep quantitatively.