  We construct a lattice model of topological order (kagome quantum spin
liquids) and solve it with unbiased quantum Monte Carlo simulations. A
three-stage anyon condensation with two transitions from a $\mathbb
Z_2\boxtimes\mathbb Z_2$ topological order to a $\mathbb Z_2$ topological order
and eventually to a trivial symmetric phase is revealed. These results provide
concrete examples of phase transitions between topological orders in quantum
magnets. The designed quantum spin liquid model and its numerical solution
offer a playground for further investigations on vestigial anyon condensation.
