Stellar toroidal magnetic fields are known to be unstable to the Tayler instability. Here we demonstrate the existence of a complementary, universal instability of toroidal magnetic fields in stratified stars with the following properties: $(i)$ it grows on the Alfv\'{e}n time $\tau_{\rm A}$; $(ii)$ once triggered, it generates differential shellular rotation about an arbitrary axis perpendicular to the symmetry axis of the magnetic field; $(iii)$ it is large-scale in the angular directions $\theta$ and $\varphi$ and develops at radial wavenumbers $k \lesssim \mathcal{N}\tau_{\rm A}/R$, where $\mathcal{N}$ is the Brunt-V$\ddot{\rm a}$is$\ddot{\rm a}$l$\ddot{\rm a}$ frequency and $R$ is the stellar radius. Thus, unlike the Tayler instability, the proposed instability is intrinsically global. Consequently, it is less susceptible to dissipative suppression than the Tayler instability and, under certain conditions, should prevail over it. This new instability may have broad implications for magnetic-field generation in stars and could modify scenarios of field formation within the Tayler-Spruit dynamo, contributing to models of efficient angular-momentum transport and associated chemical mixing in stellar interiors.