SrTiO₃ displays superconductivity, incipient ferroelectricity and can host two-dimensional electron gases (2DEGs). These phenomena occur below the cubic-to-tetragonal transition, where ferroelastic twins emerge as a result of the lower symmetry and have a strong impact on all the aforementioned properties, To shed light on these issues, we exploited the magnetoelastic coupling caused by the imprinting of ferroelastic domains into magnetostrictive films on SrTiO₃. With this approach we imaged the spatial distribution of ferroelastic twins under the action of in-situ applied electric fields and revealed the expected anisotropic dielectric behavior of tetragonal SrTiO₃. Based on first-principles and Landau theory, we associate the observed anisotropy to the emergence of an antiferroelectric (AFE) lattice instability of the Ti ions that couples to polar and AFD lattice modes. The appearance of this coupling also solves the longstanding issue of the origin of the R-point infrared-active phonon which previously was generically assigned to the back-folding of the Brillouin zone in the tetragonal phase. Our study foresees the emergence of antiferroelectric instabilities in other oxide perovskites, a prediction that deserves further research in the lattice dynamics of these materials.