This paper develops a novel multi-rate, multi-solver co-simulation framework combining dynamic phasors, transient stability, base-frequency dynamic phasors for frequency-adaptive simulation of transients, and electromagnetic transient (EMT) models. This framework subdivides a given power network into several types of subsystems based on the connected devices, required accuracy in representing dynamic details, electrical distance from perturbations, and the intended purpose of the study; as such, the paper describes methods and guidelines to simulate each subsystem using the most appropriate solver and time-step size to maximize simulation efficiency and accuracy. It also addresses the tasks of multiple interfacing and solver interactions that are essential in coupling different solvers. The proposed framework is built around an industrial-grade EMT simulator, to which other solvers are interfaced, enabling access to a variety of power system models and distinct features. The accuracy and efficiency of the framework are demonstrated through co-simulations carried out on a modified version of the 118-bus network, which includes an MMC-HVDC system.