Let \(\mathcal C\) be a norm ideal and \((A_1, \dots,A_n)\) be a commuting tuple of self-adjoint operators on a separable Hilbert space \(\mathcal H\). Continuing previous investigations from a series of articles in the same journal, the author considers the problem of determining whether or not \((A_1,\dots,A_n)\) can be simultaneously diagonalized modulo \(\mathcal C\). The author provides a spectral condition which is sufficient and which seems to be close to being necessary for a large class of norm ideals, including the Lorentz ideals. For the class of Orlicz ideals, this condition is also necessary. As an interesting open problem, the question of necessity for any norm ideal is stated.