  In this paper we complete a systematic study on quasinormal modes (QNMs) and
late time tails for scalar, Dirac and Maxwell fields on a spherically symmetric
Schwarzschild-like black hole with a global monopole in the Einstein-bumblebee
theory. To look for QNMs, we solve the equations of motion numerically by
employing both the matrix and the WKB methods, and find good agreements for
numeric data obtained by these two techniques in the regime when both are
valid. The impact of the bumblebee parameter $c$, the monopole parameter
$\eta^2$ and the multipole number $\ell$ on the fundamental QNMs is analyzed in
detail. Our results are shown in terms of the QNMs measured by $\sqrt{1+c}\,M$,
where $M$ is a black hole mass parameter. We observe, by increasing $c$
($\eta^2$) with fixed first few $\ell$, that the real part of QNMs increases
for all spin fields; while the magnitude of the imaginary part decreases for
scalar and Dirac fields but increases for Maxwell fields. By increasing the
multipole number $\ell$ with fixed other parameters, we disclose that the real
part of QNMs for all spin fields increases while the magnitude of the imaginary
part decreases for scalar and Dirac fields but increases for Maxwell fields. In
the eikonal limit, QNMs for all spin fields coincide with each other and the
real part scale linearly with $\ell$. In particular, \textit{only} the real
part of the asymptotic QNMs is dependent on the bumblebee and monopole
parameters. In addition, it is shown that the late time behavior is determined
not only by the multipole number but also by the bumblebee and monopole
parameters, and is distinct for bosonic and fermonic fields. Our results
indicate, both in the context of QNMs and late time tails, that the bumblebee
field and the monopole field play the same role in determining the dynamic
evolution of perturbation fields.
