Others have reported similar problems. It might be a bug, or it might be a limitation of how sistats and stats work together. Your best approach is to simplify your first stats operation that is working with the summary index data and do any additional computation / analysis / filtering later in the search pipeline.
I would recommend opening a support case. Splunk Support / Engineering are in the best position to determine if this is a bug or a limitation of the current summary index system.
In the meanwhile, you can probably perform a workaround similar to this:
index="client_tracking" tag::eventtype=normal_user tracking_phase=20110420_A
| sistats count by ss_action
For your populating search, and
| stats count by ss_action | search ss_action="click.button"
Actually my real stats is a lot more complex and includes several aggregation functions and a by clause, so I don't think this approach would work.
But perhaps computing an "eval" field prior to sistats and only using count(fieldname) will be ok. I'll check and post back.