  We discuss anomaly cancellation in $U(2)$ gauge theories in four dimensions.
For a $U(2)$ gauge theory defined with a spin structure, the vanishing of the
bordism group $\Omega_5^{\text{Spin}}(BU(2))$ implies that there can be no
global anomalies, in contrast to the related case of an $SU(2)$ gauge theory.
We show explicitly that the familiar $SU(2)$ global anomaly is replaced by a
local anomaly when $SU(2)$ is embedded in $U(2)$. There must be an even number
of fermions with isospin $2r+1/2$, for $r\in \mathbb{Z}_{\geq 0}$, for this
local anomaly to cancel. The case of a $U(2)$ theory defined without a choice
of spin structure but rather using a spin-$U(2)$ structure, which is possible
when all fermions (bosons) have half-integer (integer) isospin and odd (even)
$U(1)$ charge, is more subtle. We find that the recently-discovered `new
$SU(2)$ global anomaly' is also equivalent, though only at the level of the
partition function, to a perturbative anomaly in the $U(2)$ theory, which is
this time a combination of a mixed gauge anomaly with a gauge-gravity anomaly.
This perturbative anomaly vanishes if there is an even number of fermions with
isospin $4r+3/2$, for $r\in \mathbb{Z}_{\geq 0}$, recovering the condition for
cancelling the new $SU(2)$ anomaly. Alternatively, this perturbative anomaly
can be cancelled by a Wess--Zumino term, leaving a low-energy theory with a
global anomaly, which can itself be cancelled by coupling to topological
degrees of freedom.
