On the boundary of a Galton-Watson tree we can define thevisibility measureby splitting mass equally between the children of each vertex, and thebranching measureby splitting unit mass equally between all vertices in thenth generation and then lettingngo to infinity. The multifractal structure of each of these measures is well studied. In this paper we address the question of ajointmultifractal spectrum, i.e. we ask for the Hausdorff dimension of the boundary points whichsimultaneouslyhave an unusual local dimension for both these measures. The resulting two-parameter spectrum exhibits a number of surprising new features, among them the emergence of a swallowtail-shaped spectrum for the visibility measure in the presence of a nontrivial condition on the branching measure.