We study the impact of rotation on the multimessenger signals of
core-collapse supernovae (CCSNe) with the occurrence of a first-order
hadron-quark phase transition (HQPT). We simulate CCSNe with the \texttt{FLASH}
code starting from a 20~$M_\odot$ progenitor with different rotation rates, and
using the RDF equation of state from \textit{Bastian} 2021 that prescribes the
HQPT. Rotation is found to delay the onset of the HQPT and the resulting
dynamical collapse of the protocompact star (PCS) due to the centrifugal
support. All models with the HQPT experience a second bounce shock which leads
to a successful explosion. The oblate PCS as deformed by rotation gives rise to
strong gravitational-wave (GW) emission around the second bounce with a peak
amplitude larger by a factor of $\sim10$ than that around the first bounce. The
breakout of the second bounce shock at the neutrinosphere produces a
$\bar{\nu}_e$-rich neutrino burst with a luminosity of serveral
10$^{53}$~erg~s$^{-1}$. In rapidly rotating models the PCS pulsation following
the second bounce generates oscillations in the neutrino signal after the
burst. In the fastest rotating model with the HQPT, a clear correlation is
found between the oscillations in the GW and neutrino signals immediately after
the second bounce. In addition, the HQPT-induced collapse leads to a jump in
the ratio of rotational kinetic energy to gravitational energy ($\beta$) of the
PCS, for which persistent GW emission may arise due to secular nonaxisymmetric
instabilities.
