  Let $M$ be a $3$-manifold with connected non-vacuos boundary which is not
spherical. Assume that $N$ is another $3$-manifold with vacuous boundary and
$N^{\ast}$ is the $3$-manifold obtained by removing from $N$ the interior of a
$3$-cell. In the present paper, we prove that the twisted Reidemeister torsion
of the connected sum $M\# N$ can be expressed as the product of the torsions of
$M$ and $N^{\ast}.$ Indeed, we show that the twisted Reidemeister torsion has a
multiplicative property on the connected sum of compact $3$-manifolds.
