We study the Heisenberg XXX spin chain with twisted periodic boundary conditions with a defective twist matrix. Using quantum separation of variables, we show that the generalised eigenvectors of this system can be characterised in terms of a modified TQ equation that we have named “Jordanian TQ equation”. We also show that the wavefunction in the separated basis is only separable for true eigenvectors, while the wavefunction for generalised eigenvectors of higher rank is given instead by a sum over weak compositions.