Role: Scribe
It helps to go into each workshop with a basic template of what you expect to see. For example, for each of the defect reduction topics I had a basic template in mind that had the following categories: data supporting the point, data contradicting the point, extensions (for when someone says, "I can support that point but only in the case of systems of type x"), and definitions (for when someone says, "but we have to define what we mean by x before I can say whether I agree with your statement"). I made sure that before we started, the categories in the template were listed on the whiteboard - thus as statements come in from the chat you can just drop them in the relevant place on the whiteboard rather than worrying about giving a full description of how it connects to the rest of the argument.

That's not a complete solution - you have to be able to change your categorization on the fly one you see what people are willing & able to talk about, and it's never exactly the same from one topic to the next - but it does help deal with the volume of information you see during the chat.