  We carry out lattice simulations of two-color QCD and spectroscopy at finite
density with two flavors of rooted-staggered quarks and a diquark source term.
As in a previous four-flavor study, for small values of the inverse gauge
coupling we observe a Goldstone spectrum which reflects the symmetry-breaking
pattern of a Gaussian symplectic chiral random-matrix ensemble (GSE) with Dyson
index $\beta_D=4$, which corresponds to any-color QCD with adjoint quarks in
the continuum instead of QC$_2$D wih fundamental quarks. We show that this
unphysical behavior occurs only inside of the bulk phase of $SU(2)$ gauge
theory, where the density of $Z_2$ monopoles is high. Using an improved gauge
action and a somewhat larger inverse coupling to suppress these monopoles, we
demonstrate that the continuum Goldstone spectrum of two-color QCD,
corresponding to a Gaussian orthogonal ensemble (GOE) with Dyson index
$\beta_D=1$, is recovered also with rooted-staggered quarks once simulations
are performed away from the bulk phase. We further demonstrate how this change
of random-matrix ensemble is reflected in the distribution of eigenvalues of
the Dirac operator. By computing the unfolded level spacings inside and outside
of the bulk phase, we demonstrate that, starting with the low-lying eigenmodes
which determine the infrared physics, the distribution of eigenmodes
continuously changes from the GSE to the GOE one as monopoles are suppressed.
