I've been especially lazy on the bus this week and not much livelier at home, so no progress to report this week. Instead here is a rather low quality photo of the tea dyed string and the un-dyed versions of the thread.
The tea dyed sections (the vertically wound bits) were left in a mug of tea with a bit of soymilk for a day, then taken out to dry without rinsing. Even in this poor photo, I think you can make out how the pink is tinged brown while the darker green is mostly the same colour but without the slight glossiness. The pale green is also tea dyed, but I couldn't remember which lot of embroidery thread I snipped it from, so there's no "before" to compare it to.
The tea dye seems to just sort of dull the colours a bit, and that makes the string work better with the shirt fabric. Or, it would work better with the fabric, if I ever get around to embroidering with it. Maybe next week.