We propose a design strategy -based on the coupling of spins, optical phonons, and strain -for systems in which magnetic (electric) phase control can be achieved by an applied electric (magnetic) field. Using first-principles density-functional theory calculations, we present a realization of this strategy for the magnetic perovskite EuTiO3.There is great interest in multiferroic materials in which ferroelectric (FE) and ferromagnetic (FM) ordering not only coexist, but in which the electrical polarization P and the magnetization M are large and strongly coupled [1,2]. One challenge in identifying strongly coupled FM-FE's that has received considerable attention in the past [3] is the scarcity of such materials in nature, as most insulators (a requirement for ferroelectricity) are paraelectric (PE) and antiferromagnetic (AFM). With the recent advances in first-principles density-functional methods for predicting [4] and in novel synthetic techniques for growing new FM-FE multiferroics, the focus has now turned to how to produce a strong coupling between the two distinct order parameters [5]. Recently, attention has focused on a mechanism in which magnetic order itself breaks inversion symmetry [6,7,8,9]. Based on this, remarkable control of the FE state by an applied magnetic field has been demonstrated in some rare-earth manganites [10,11], however, the natural scale of the spontaneous polarization thus induced is very small, of the order of nC/cm 2 . Furthermore, the magnetic state appears to be rather insensitive to an applied electric field for this class of materials.As a result, it is clearly advantageous to explore other possible mechanisms for strongly coupled multiferroism. A fruitful starting point for identifying such mechanisms is the observation, recently discussed by Tokura [5], that the basic physics of a strong M-P coupling involves a competition between different ordered states, e.g. between a FM-FE state and an AFM-PE state. In this Letter, we present a new approach for designing a strongly coupled multiferroic in which the interplay of spins, optical phonons, and strain leads to such a competition.The criteria that a system must satisfy for this proposed mechanism to be realized are as follows: (1) It must be an AFM-PE insulator in which at least one infrared-active (ir) phonon is coupled to the magnetic order, (2) the spins in the AFM ground state should align with the application of a magnetic field of modest strength, (3) this alignment should decrease the frequency of the spin-coupled ir-active phonon, and, (4) the key to our approach, the ir-active mode of interest must be strongly coupled to strain. Epitaxial strain can have profound effects on the properties of thin films [12]. In our design strategy we use epitaxial strain to dial into the region of the phase diagram where a spin-phonondriven destabilization of the lattice actually occurs. The FM-FE phase thus produced is a low-lying state competing with the AFM-PE ground state. As a direct result of this competition, magnetic and elec...