Hello -
Thanks for your reply. We did run some tracing on the client and server. The pattern we're seeing is that the request gets to IIS fairly quickly, then IIS takes 2-7 seconds to serve the tile. We have fiddler running on the ArcGIS Server too, which shows the web adapter request to ArcGIS Server, and the tile fetching as a whole is very quick. So there's something in IIS that is a sticking point, which could be related to the web adaptor.
This only happens occasionally, maybe 10 minutes out of an hour. Let me know if this may be related to anything you've seen.
Thanks!
-Brian