  We uncover a new type of turbulence -- activity-induced homogeneous and
isotropic turbulence in a model that has been employed to investigate
motility-induced phase separation (MIPS) in a system of microswimmers. The
active Cahn-Hilliard-Navier-Stokes equations (CHNS), also called active model
H, provides a natural theoretical framework for our study. In this CHNS model,
a single scalar order parameter $\phi$, positive (negative) in regions of high
(low) microswimmer density, is coupled with the velocity field $\bm u$. The
activity of the microswimmers is governed by an activity parameter $\zeta$ that
is positive for \textit{extensile} swimmers and negative for
\textit{contractile} swimmers. With extensile swimmers, this system undergoes
complete phase separation, which is similar to that in binary-fluid mixtures.
By carrying out pseudospectral direct numerical simulations (DNSs), we show,
for the first time, that this model (a) develops an emergent nonequilibrium,
but statistically steady, state (NESS) of active turbulence, for the case of
contractile swimmers, if $\zeta$ is sufficiently large and negative and (b)
this turbulence arrests the phase separation. We quantify this suppression by
showing how the coarsening-arrest length scale does not grow indefinitely, with
time $t$, but saturates at a finite value at large times. We characterise the
statistical properties of this active-scalar turbulence by employing energy
spectra and fluxes and the spectrum of $\phi$. For sufficiently high Reynolds
numbers, the energy spectrum $\mathcal E(k)$ displays an inertial range, with a
power-law dependence on the wavenumber $k$. We demonstrate that, in this range,
the flux $\Pi(k)$ assumes a nearly constant, negative value, which indicates
that the system shows an inverse cascade of energy, even though energy
injection occurs over a wide range of wavenumbers in our active-CHNS model.
