  We study the renormalized analytic torsion of complete manifolds with fibred
boundary metrics, also referred to as $\phi$-metrics. We establish invariance
of the torsion under suitable deformations of the metric, and establish a
gluing formula. As an application, we relate the analytic torsions for complete
$\phi$- and incomplete wedge-metrics. As a simple consequence we recover a
result by Sher and Guillarmou about analytic torsion under conic degeneration.
