  This paper studies a Shannon-theoretic version of the generalized
distribution preserving quantization problem where a stationary and memoryless
source is encoded subject to a distortion constraint and the additional
requirement that the reproduction also be stationary and memoryless with a
given distribution. The encoder and decoder are stochastic and assumed to have
access to independent common randomness. Recent work has characterized the
minimum achievable coding rate at a given distortion level when unlimited
common randomness is available. Here we consider the general case where the
available common randomness may be rate limited. Our main result completely
characterizes the set of achievable coding and common randomness rate pairs at
any distortion level, thereby providing the optimal tradeoff between these two
rate quantities. We also consider two variations of this problem where we
investigate the effect of relaxing the strict output distribution constraint
and the role of `private randomness' used by the decoder on the rate region.
Our results have strong connections with Cuff's recent work on distributed
channel synthesis. In particular, our achievability proof combines a coupling
argument with the approach developed by Cuff, where instead of explicitly
constructing the encoder-decoder pair, a joint distribution is constructed from
which a desired encoder-decoder pair is established. We show however that for
our problem, the separated solution of first finding an optimal channel and
then synthesizing this channel results in a suboptimal rate region.
