  The quantum anomaly of a global symmetry is known to strongly constrain the
allowed low-energy physics in a clean and isolated quantum system. However, the
effect of quantum anomalies in disordered systems is much less understood,
especially when the global symmetry is only preserved on average by the
disorder. In this work, we focus on disordered systems with both average and
exact symmetries $A\times K$, where the exact symmetry $K$ is respected in
every disorder configuration, and the average $A$ is only preserved on average
by the disorder ensemble. When there is a mixed quantum anomaly between the
average and exact symmetries, we argue that the mixed state representing the
ensemble of disordered ground states cannot be featureless. While disordered
mixed states smoothly connected to the anomaly-compatible phases in clean limit
are certainly allowed, we also found disordered phases that have no clean-limit
counterparts, including the glassy states with strong-to-weak symmetry
breaking, and average topological orders for certain anomalies. We construct
solvable lattice models to demonstrate each of these possibilities. We also
provide a field-theoretic argument to provide a criterion for whether a given
average-exact mixed anomaly admits a compatible average topological order.
