Interesting. Let me try to see if I understand and correct me where I get it wrong.
You are interested in a posterior distribution p(\theta) and assume a variational approximation q(\theta | \phi) but instead of directly optimizing KL(p || q), you want to construct a Markov chain \phi^{(0)} , \phi^{(1)},... such that the draws of \phi^{(l)} lead to a good variational approximation q(\theta | \phi^{(l)}). Is that right? If so, what does the transition kernel for \phi | \phi' look like?
Do you have any references / reading material that could help me understand this?