  Recently a Born-Infeld action for dark energy and dark matter that uses
additional affine connections was proposed. At background level, it was shown
that the new proposal can mimic the standard cosmological evolution. In Bianchi
cosmologies, contrary to the scalar field approach (e.g., Chaplygin gas), the
new approach leads to anisotropic pressure, raising the issues of stability of
the isotropic solution under anisotropic perturbations and, being it stable,
how the anisotropies evolve. In this work, the Eddington-Born-Infeld proposal
is extended to a Bianchi type I scenario and residual post-inflationary
anisotropies are shown to decay in time. Moreover, it is shown that the shears
decay following a damped oscillatory pattern, instead of the standard
exponential-like decay. Allowing for some fine tuning on the initial
conditions, standard theoretical bounds on the shears can be avoided.
