We present a new kind of self-consistent GW approximation based on the all-electron, full-potential linear muffin-tin orbital method. By iterating the eigenfunctions of the GW Hamiltonian, self-consistency in both the charge density and the quasiparticle spectrum is achieved. We explain why this form of self-consistency should be preferred to the conventional one. Some results for Si (a representative semiconductor) are presented. Finally we consider many details in the electronic structure of the antiferromagnetic insulators MnO and NiO. Excellent agreement with experiment is shown for many properties, suggesting that a Landau quasiparticle (energy band) picture provides a reasonable description of electronic structure even in these correlated materials.