  Density perturbations in cosmology, i.e. spherically symmetric adiabatic
perturbations of a Friedmann-Lema\^itre-Robertson-Walker (FLRW) spacetime, are
locally exactly equivalent to a different FLRW solution, as long as their
wavelength is much larger than the sound horizon of all fluid components. This
fact is known as the "separate universe" paradigm. However, no such relation is
known for anisotropic adiabatic perturbations, which correspond to an FLRW
spacetime with large-scale tidal fields. Here, we provide a closed, fully
relativistic set of evolutionary equations for the nonlinear evolution of such
modes, based on the conformal Fermi (CFC) frame. We show explicitly that the
tidal effects are encoded by the Weyl tensor, and are hence entirely different
from an anisotropic Bianchi I spacetime, where the anisotropy is sourced by the
Ricci tensor. In order to close the system, certain higher derivative terms
have to be dropped. We show that this approximation is equivalent to the local
tidal approximation of Hui and Bertschinger (1995). We also show that this very
simple set of equations matches the exact evolution of the density field at
second order, but fails at third and higher order. This provides a useful,
easy-to-use framework for computing the fully relativistic growth of structure
at second order.
