  We consider the semilinear wave equation $\Box_g u+a u^4=0$, $a\neq 0$, on a
Lorentzian manifold $(M,g)$ with timelike boundary. We show that from the
knowledge of the Dirichlet-to-Neumann map one can recover the metric $g$ and
the coefficient $a$ up to natural obstructions. Our proof rests on the analysis
of the interaction of distorted plane waves together with a scattering control
argument, as well as Gaussian beam solutions.
