Multiblock copolymers (MBCPs) comprising polyester and polyacrylate segments offer an efficient strategy for enhancing the performance of polyester and polyolefin blends but synthesis and structural modification of these MBCPs remains challenging. Here, we propose a method for synthesizing MBCPs via the switchable polymerization of epoxides, cyclic anhydrides, and acrylates using a dinuclear Co-complex, wherein the anhydride acts as a switcher. Detailed studies on the copolymerization process reveal that the successful synthesis of MBCPs is achieved by intramolecular bimetallic synergistic catalysis, producing MBCPs with controlled molecular weights and narrow dispersities. Owing to the high compatibility of the monomers, this method allows for producing MBCPs with diverse structures and block numbers. Moreover, the resulting MBCPs effectively enhance the performance of the polyester and polyacrylate blends, improving the toughness of polyesters. Studies on microphase separation show that MBCPs can effectively compatibilize immiscible blends, highlighting their potential as compatibilizers.