Three-dimensional antiferromagnets with random magnetic anisotropy (RMA) that have been experimentally studied to date have competing two-dimensional and three-dimensional exchange interactions which can obscure the authentic effects of RMA. The magnetic phase diagram of Fe_{x}Ni_{1-x}F_{2} epitaxial thin films with true random single-ion anisotropy was deduced from magnetometry and neutron scattering measurements and analyzed using mean-field theory. Regions with uniaxial, oblique, and easy-plane anisotropies were identified. A RMA-induced glass region was discovered where a Griffiths-like breakdown of long-range spin order occurs.