  Bilayer quantum Hall systems, realized either in two separated wells or in
the lowest two sub-bands of a wide quantum well, provide an experimentally
realizable way to tune between competing quantum orders at the same filling
fraction. Using newly developed density matrix renormalization group techniques
combined with exact diagonalization, we return to the problem of quantum Hall
bilayers at filling $\nu = 1/3 + 1/3$. We first consider the Coulomb
interaction at bilayer separation $d$, bilayer tunneling energy
$\Delta_\textrm{SAS}$, and individual layer width $w$, where we find a phase
diagram which includes three competing Abelian phases: a bilayer-Laughlin phase
(two nearly decoupled $\nu = 1/3$ layers); a bilayer-spin singlet phase; and a
bilayer-symmetric phase. We also study the order of the transitions between
these phases. A variety of non-Abelian phases have also been proposed for these
systems. While absent in the simplest phase diagram, by slightly modifying the
interlayer repulsion we find a robust non-Abelian phase which we identify as
the "interlayer-Pfaffian" phase. In addition to non-Abelian statistics similar
to the Moore-Read state, it exhibits a novel form of bilayer-spin charge
separation. Our results suggest that $\nu = 1/3 + 1/3$ systems merit further
experimental study.
