It turns out our problem was caused by a WAN optimization device that was excluding all traffic to/from the core. I suspect the pass-thru rule was somehow corrupting the AMT traffic, because it wasn't blocked, it just wasn't correct. Removing this rule and optimizing all traffic to/from the core seems to have resolved this issue.
We found that when our River Dead was active between our cores and other devices everything slowed to a crawl. Imagine the Console 50% slower