  Controlling the contributions from doubly Cabibbo-suppressed penguin
topologies in the decays $B_d^0\to J/\psi K^0_S$ and $B_s^0\to J/\psi\phi$ is
mandatory to reach the highest possible precision in the measurement of the
$B^0_q$--$\bar B^0_q$ ($q=d,s$) mixing phases $\phi_d$ and $\phi_s$. The
penguin contributions can be determined using a strategy based on the $SU(3)$
flavour symmetry of QCD. Using the latest experimental data, we update our
combined analysis of the decays $B_d^0\to J/\psi K^0_S$, $B_s^0\to J/\psi\phi$
and their control channels $B_s^0\to J/\psi K^0_S$, $B_d^0\to J/\psi \pi^0$ and
$B_d^0\to J/\psi \rho^0$. This allows us to simultaneously determine the
penguin parameters and both mixing phases. We discuss how the branching
fractions of these decays can be used to probe the size of non-factorisable
$SU(3)$-breaking effects, which form the main theoretical uncertainty
associated with our $SU(3)$-based strategy, and provide new insights into the
factorisation approach.
