  In an irrotational dust universe, the locally free gravitational field is
covariantly described by the gravito-electric and gravito-magnetic tensors
$E_{ab}$ and $H_{ab}$. In Newtonian theory, $H_{ab}=0$ and $E_{ab}$ is the
tidal tensor. Newtonian-like dust universes in general relativity (i.e. with
$H_{ab}=0$, often called `silent') have been shown to be inconsistent in
general and unlikely to extend beyond the known spatially homogeneous or
Szekeres examples. Furthermore, they are subject to a linearization
instability. Here we show that `anti-Newtonian' universes, i.e. with purely
gravito-magnetic field, so that $E_{ab} = 0\neq H_{ab}$, are also subject to
severe integrability conditions. Thus these models are inconsistent in general.
We show also that there are no anti-Newtonian spacetimes that are linearized
perturbations of Robertson-Walker universes. The only $E_{ab}=0\neq H_{ab}$
solution known to us is not a dust solution, and we show that it is
kinematically G\"{o}del-like but dynamically unphysical.
