Anisotropic stresses are ubiquitous in nature, but their modeling in general relativity is poorly understood and frame dependent. We introduce the first study on the dynamical properties of anisotropic self-gravitating fluids in a covariant framework. Our description is particularly useful in the context of tests of the black hole paradigm, wherein ultracompact objects are used as black hole mimickers but otherwise lack a proper theoretical framework. We show the following: (i) anisotropic stars can be as compact and as massive as black holes, even for very small anisotropy parameters; (ii) the nonlinear dynamics of the 1 + 1 system is in good agreement with linearized calculations, and shows that configurations below the maximum mass are nonlinearly stable; (iii) strongly anisotropic stars have vanishing tidal Love numbers in the black-hole limit; and (iv) their formation will usually be accompanied by gravitational-wave echoes at late times.