  We present an effective Eulerian description, in the non-relativistic regime,
of the growth of cosmological perturbations around a homogeneous but
anisotropic Bianchi I spacetime background. We assume a small deviation from
isotropy, sourced at late times for example by dark energy anisotropic stress.
We thus derive an analytic solution for the linear dark matter density
contrast, and use it in a formal perturbative approach which allows us to
derive a second order (non-linear) solution. As an application of the procedure
followed here we derive analytic expressions for the power spectrum and the
bispectrum of the dark matter density contrast. The power spectrum receives a
quadrupolar correction as expected, and the bispectrum receives several
angle-dependent corrections. Quite generally, we find that the contribution of
a late-time phase of anisotropic expansion to the growth of structure peaks at
a finite redshift between CMB decoupling and today, tough the exact redshift
value is model-dependent.
