  We argue that strings (vortices) and monopoles are confined, when fields
receiving nontrivial Aharonov-Bohm (AB) phases around a string develop vacuum
expectation values (VEVs). We illustrate this in an $SU(2) \times U(1)$ gauge
theory with charged triplet complex scalar fields admitting Alice strings and
monopoles, by introducing charged doublet scalar fields receiving nontrivial AB
phases around the Alice string. The Alice string carries a half $U(1)$ magnetic
flux and $1/4$ $SU(2)$ magnetic flux taking a value in two of the $SU(2)$
generators characterizing the $U(1)$ modulus. This string is not confined in
the absence of a doublet VEV in the sense that the $SU(2)$ magnetic flux can be
detected at large distance by an AB phase around the string. When the doublet
field develops VEVs, there appear two kinds of phases that we call deconfined
and confined phases. When a single Alice string is present in the deconfined
phase, the $U(1)$ modulus of the string and the vacuum moduli are locked (the
bulk-soliton moduli locking). In the confined phase, the Alice string is
inevitably attached by a domain wall that we call an AB defect and is confined
with an anti-Alice string or another Alice string with the same $SU(2)$ flux.
Depending on the partner, the pair annihilates or forms a stable doubly-wound
Alice string having an $SU(2)$ magnetic flux inside the core, whose color
cannot be detected at large distance by AB phases, implying the "color"
confinement. The theory also admits stable Abrikosov-Nielsen-Olesen string and
a ${\mathbb Z}_2$ string in the absence of the doublet VEVs, and each decays
into two Alice strings in the presence of the doublet VEVs. A monopole in this
theory can be constructed as a closed Alice string with the $U(1)$ modulus
twisted once, and we show that with the doublet VEVs, monopoles are also
confined to monopole mesons of the monopole charge two.
