  The magnetic, noncollinear parametrization of Dudarev's DFT+$U$ method is
generalized to fully-relativistic ultrasoft pseudopotentials. We present the
definition of the DFT+$U$ total energy functional, and the calculation of
forces and stresses in the case of orthogonalized atomic orbitals defining the
localised Hubbard manifold, where additional contributions arising from the
derivative of the inverse square root of the overlap matrix appear. We further
extend the perturbative calculation of the Hubbard $U$ parameters within
density-functional perturbation theory to the noncollinear relativistic case,
by exploiting an existing and recently developed theoretical approach that
takes advantage of the time-reversal operator to solve a second Sternheimer
equation. We validate and apply the new scheme by studying the electronic
structure and the thermodynamics of the binary compounds EuX (where X = O, S,
Se, Te is a chalcogen atom), as representative simple crystals, and of the
pyrochlore Cd$_2$Os$_2$O$_7$, representative of a more structurally complex
oxide.
