  We propose a new Monte Carlo algorithm for complex discrete distributions.
The algorithm is motivated by the N-Fold Way, which is an ingenious
event-driven MCMC sampler that avoids rejection moves at any specific state.
The N-Fold Way can however get "trapped" in cycles. We surmount this problem by
modifying the sampling process. This correction does introduce bias, but the
bias is subsequently corrected with a carefully engineered importance sampler.
