  We study the impact of quenched disorder (random exchange couplings or site
dilution) on easy-plane pyrochlore antiferromagnets. In the clean system,
order-by-disorder selects a magnetically ordered state from a classically
degenerate manifold. In the presence of randomness, however, different orders
can be chosen locally depending on details of the disorder configuration. Using
a combination of analytical considerations and classical Monte-Carlo
simulations, we argue that any long-range-ordered magnetic state is destroyed
beyond a critical level of randomness where the system breaks into magnetic
domains due to random exchange anisotropies, becoming, therefore, a glass of
spin clusters, in accordance with the available experimental data. These random
anisotropies originate from off-diagonal exchange couplings in the microscopic
Hamiltonian, establishing their relevance to other magnets with strong
spin-orbit coupling.
