It is shown that the accepted proof of the anti-symmetrised geminal power (AGP) wave functions lack of size consistency is not general enough to constitute a proof for the size consistency of the AGP wave function. The origin of the size consistency problem for AGP wave function in previous proofs is shown to stem from the perceived notion that the natural orbitals of the AGP always can be localised or guessed a priori. We here show that by applying different constraints on a more general geminal coefficient matrix that the ionised/electron-attached determinants can be eliminated in different ways in a spin-restricted basis, which is not possible in the accepted proof. Furthermore it is shown how different constraints on the coefficients in the geminal coefficient matrix can lead to different ionisation channels upon dissociation. We discuss the consequences of the generation of natural orbitals from the solution of the AGP using a more general coefficient matrix. Finally the modern use of the natural AGP as a reference function for another correlation method is discussed where improvements to the orbitals used in the modern AGP are suggested.