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The problem was a side effect of some code that was blocking. I managed to detect the blocking code using the eventlet option described in this article.
There were 2 places where blocking was occuring: DNS lookups, and MySQL database access. I managed to resolve the first by installing the dnspython package, and the second my using the undocumented MySQLdb option in eventlet:
import eventlet
eventlet.monkey_patch()
eventlet.monkey_patch(MySQLdb=True)
Source:stackexchange.com