  A Bianchi IX Mixmaster spacetime is the most general spatially homogeneous
solution of Einstein's equations and it can represent the space-averaged
Universe. We introduce two novel mechanisms resulting in a Mixmaster Universe
with non-singular bounces which are quasi-isotropic. A fluid with a non-linear
equation of state allows non-singular bounces. Using negative anisotropic
stresses successfully isotropises this Universe and mitigates the well known
Mixmaster chaotic behaviour. Thus the Universe can be an eternal Mixmaster,
going through an infinite series of different cycles separated by bounces, with
a sizable fraction of cycles isotropic enough to be well approximated by a
standard Friedmann-Lema\^itre-Robertson-Walker model from the radiation era
onward.
