If the eagle was a direct embroidery I would almost think that it would have to be done before the tunic was assembled , or at very least the sleeve would have been taken apart to allow the work. These eagles were not embroidered per se, but rather executed in gold wire requiring 100's of small pieces to be sewn down. Nearly, if not actually, impossible on an assembled tunic sleeve. Small quip here, that doesn't take away from the eagle in any way.

Regards,
William Unland