  We study theoretically the magic-angle twisted bilayer graphene with
proximity-induced Ising and Rashba spin-orbit couplings on the top layer.
Topological flat bands (with three distinct phases) are generically realized by
the spin-orbit couplings. Using a mean field analysis, we find that (partial)
valley polarization prevails for a wide range of doping, suppressing the usual
superconductivity with a pairing between time-reversal partners. Remarkably, we
uncover that observable unconventional intervalley interband phonon-mediated
superconductivity (with the highest $T_c\approx 1.2$K) can coexist with strong
valley imbalance due to the approximate Fermi surface nesting between two flat
bands not related by time-reversal symmetry, and the dominant pairing is an
intersublattice Ising pairing, corresponding to a mixture of $p$- and
$d$-waves. In contrast, the intrasublattice Ising phonon-mediated
superconductivity with $s$- and $f$-wave mixing emerges in the absence of
valley imbalance. Our work reveals an unprecedented route of realizing
unconventional superconductivity.
