  By combining the nonconserved spin-flip dynamics driving ferromagnetic
ordering with the conserved Kawasaki-exchange dynamics driving phase
segregation, we perform Monte Carlo simulations of the nearest neighbor Ising
model. Such a set up mimics a system consisting of a binary mixture of
\emph{isomers} which is simultaneously undergoing a segregation and an
\emph{interconversion} reaction among themselves . Here, we study such a system
following a quench from the high-temperature homogeneous phase to a temperature
below the demixing transition. We monitor the growth of domains of both the
\emph{winner}, the \emph{isomer} which survives as the majority and the
\emph{loser}, the \emph{isomer} that perishes. Our results show a strong
interplay of the two dynamics at early times leading to a growth of the average
domain size of both the \emph{winner} and \emph{loser} as $\sim t^{1/7}$,
slower than a purely phase-segregating system. At later times, eventually the
dynamics becomes reaction dominated, and the \emph{winner} exhibits a $\sim
t^{1/2}$ growth, expected for a system with purely nonconserved dynamics. On
the other hand, the \emph{loser} at first show a faster growth, albeit, slower
than the \emph{winner}, and then starts to decay before it almost vanishes.
Further, we estimate the time $\tau_s$ marking the crossover from the
early-time slow growth to the late-time reaction dominated faster growth. As a
function of the reaction probability $p_r$, we observe a power-law scaling
$\tau_s \sim p_r^{-x}$, where $x\approx 1.05$, irrespective of temperature. For
a fixed value of $p_r$ too, $\tau_s$ appears to be independent of temperature.
