  Markov chain Monte Carlo is a widely-used technique for generating a
dependent sequence of samples from complex distributions. Conventionally, these
methods require a source of independent random variates. Most implementations
use pseudo-random numbers instead because generating true independent variates
with a physical system is not straightforward. In this paper we show how to
modify some commonly used Markov chains to use a dependent stream of random
numbers in place of independent uniform variates. The resulting Markov chains
have the correct invariant distribution without requiring detailed knowledge of
the stream's dependencies or even its marginal distribution. As a side-effect,
sometimes far fewer random numbers are required to obtain accurate results.
