In the context of a special class of tensor-multi-scalar theories of gravity for which the target-space metric admits an isometry under which the theory is invariant, we present rotating vacuum solutions. When regularity is imposed, the primary scalar fields behave as nontopological solitons, which can support the existence of additional topological fields. In the presence of an null hypersurface the solutions are hairy black holes. The global quantities deviate from the ones given by the usual boson stars and the KBSH as the Gauss curvature varies. Non-invariant quantities such as radius, horizon area, and compactness, depend on how these extra gravitational degrees of freedom couple to ordinary matter.