Linear Donati compatibility conditions guarantee that the components of symmetric tensor fields are those of linearized change of metric or linearized change of curvature tensor fields associated with the displacement vector field arising in a linearly elastic structure when it is subjected to applied forces. These compatibility conditions take the form of variational equations with divergence-free tensor fields as test-functions, by contrast with Saint-Venant compatibility conditions, which take the form of systems of partial differential equations. In this paper, we identify and justify nonlinear Donati compatibility conditions that apply to a nonlinearly elastic plate modeled by the Kirchhoff-von K'arm'an-Love theory.