  We investigate numerically the behaviour of a phase-separating mixture of a
blue phase I liquid crystal with an isotropic fluid. The resulting morphology
is primarily controlled by an inverse capillary number, $\chi$, setting the
balance between interfacial and elastic forces. When $\chi$ and the
concentration of the isotropic component are both low, the blue phase
disclination lattice templates a cubic array of fluid cylinders. For larger
$\chi$, the isotropic phase arranges primarily into liquid emulsion droplets
which coarsen very slowly, rewiring the blue phase disclination lines into an
amorphous elastic network. Our blue phase/simple fluid composites can be
externally manipulated: an electric field can trigger a morphological
transition between cubic fluid cylinder phases with different topologies.
