  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
that: (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;
(iv) their formation will usually be accompanied by gravitational-wave echoes
at late times.
