High harmonic radiation is produced when atoms or molecules are ionized by an intense femtosecond laser pulse. The radiated spectrum has been shown experimentally to contain information on the electronic structure of the molecule, which can be interpreted as an image of a single molecular orbital. Previous theory for high harmonic generation has been limited to the single-active-electron approximation. Utilizing semisudden approximation, the authors develop a theory of the recombination step in high harmonic generation and tomographic reconstruction in multielectron systems, taking into account electron spin statistics and electron-electron correlations within the parent molecule and the ion. They show that the resulting corrections significantly modify the theoretical predictions, and bring them in a better agreement with experiment. They further show that exchange contributions to harmonic radiation can be used to extract additional information on the electronic wave function.