  We use a continuum, two-fluid approach to study a mixture of two active
nematic fluids. Even in the absence of thermodynamically-driven ordering, for
mixtures of different activities we observe turbulent microphase separation,
where domains form and disintegrate chaotically in an active turbulent
background. This is a weak effect if there is no elastic nematic alignment
between the two fluid components, but is greatly enhanced in the presence of an
elastic alignment or substrate friction. We interpret the results in terms of
relative flows between the two species which result from active anchoring at
concentration gradients. Our results may have relevance in interpreting
epithelial cell sorting and the dynamics of multi-species bacterial colonies.
