  $G_2$-QCD, in which the exceptional Lie group $G_2$ replaces the $SU(3)$
gauge group of QCD, does not suffer from a fermion sign problem. It can
therefore be simulated also at comparatively low temperatures and high
densities on the lattice, which hence allows to map out the phase diagram of
this QCD-like theory. We briefly review some of our previous results from such
finite density simulations to then present further evidence for a first-order
transition to what might be called $G_2$-nuclear matter. In order to isolate
diquark condensation effects, we introduce simulations with Majorana fermions
and diquark sources. This allows to disentangle states in the spectrum that are
connected by charge conjugation. We discuss chiral symmetry in the presence of
diquark sources and present first results from our ongoing large-scale
simulations.
