We have studied the asymptotic regularity of solutions of Hartree-Fock equations for Coulomb systems. Fock operators are discussed within the calculus of pseudo-differential operators on conical manifolds. First, the non-self-consistent-field case is considered which means that the functions that enter into the nonlinear terms are not the eigenfunctions of the Fock operator itself. We introduce asymptotic regularity conditions on the functions that build up the Fock operator which guaranty ellipticity on the open stretched cone $\mathbb{R}_+ \times S^2$. This proves existence of a parametrix and corresponding smoothing operator from which it follows that the eigenfunctions of the Fock operator belong to a certain class of weighted Sobolev spaces with prescribed asymptotics. A refined analysis actually demonstrates that the eigenfunctions are precisely of the form required for ellipticity. Using a fixed-point argument based on Canc{\`e}s and Le Bris analysis of the level-shift algorithm, we can show that self-consistent-field solutions of Hartree-Fock equations with prescribed asymptotics exist.