  Quark-hadron continuity with two-flavor quarks that was proposed recently
connects hadronic matter with neutron $^3P_2$ superfluidity and two-flavor
dense quark matter. This two-flavor dense quark phase consists of the
coexistence of the 2SC condensates and the $P$-wave diquark condensates of
$d$-quarks, which gives rise to color superconductivity as well as
superfluidity. We classify vortices in this phase. The most stable vortices are
what we call the non-Abelian Alice strings, which are superfluid vortices with
non-Abelian color magnetic fluxes therein, exhibiting so-called topological
obstruction, or a non-Abelian generalization of the Alice property. We show
that a single Abelian superfluid vortex is unstable against decay into three
non-Abelian Alice strings. We discover that a non-Abelian Alice string carries
orientational moduli of the real projective space $\mathbb{R}P^2$ corresponding
to the color flux therein in the presence of the $P$-wave condensates alone. We
calculate Aharanov-Bohm (AB) phases around the non-Abelian Alice string, and
find that the 2SC condensates and string's orientational moduli must be aligned
with each other because of single-valuedness of the AB phases of the 2SC
condensates.
