  Let $K$ be the connected sum of knots $K_1,\ldots,K_n$. It is known that the
$\mathrm{SL}_2(\mathbb{C})$-character variety of the knot exterior of $K$ has a
component of dimension $\geq 2$ as the connected sum admits a so-called
bending. We show that there is a natural way to define the adjoint Reidemeister
torsion for such a high-dimensional component and prove that it is locally
constant on a subset of the character variety where the trace of a meridian is
constant. We also prove that the adjoint Reidemeister torsion of $K$ satisfies
the vanishing identity if each $K_i$ does so.
