By ab initio many-body quantum chemistry calculations, we determine the strength of the symmetric anisotropy in the 5d⁵ j ≈ 1/2 layered material Ba₂IrO₄. While the calculated anisotropic couplings come out in the range of a few meV, orders of magnitude stronger than in analogous 3d transition-metal compounds, the Heisenberg superexchange still defines the largest energy scale. The ab initio results reveal that individual layers of Ba₂IrO₄ provide a close realization of the quantum spin-1/2 Heisenberg-compass model on the square lattice. We show that the experimentally observed basal-plane antiferromagnetism can be accounted for by including additional interlayer interactions and the associated order-by-disorder quantum-mechanical effects, in analogy to undoped layered cuprates.