So after a few days of waiting, I get the following email from Network Solutions.
Let's look more closely at that authorization code. Yeah, normally that's the sort of thing that you should never share.. but:
Oh wait, I didn't read this line in the email:
If you are planning to transfer your domain to another registrar, we would like to do whatever it takes to keep your business - please let us know how we can improve our service to you.Presumably this a way of doing whatever it takes. I did even drill down into the HTML source to make sure it wasn't my mail client screwing up. It seems that I'm not the only person who has had problems transferring their domain out according to this story.
UPDATE 2014-10-03: I raised a ticket which was acknowledged.. and then ignored completely. NetSol are breaking ICANN regulations by not providing the authorisation code in a timely manner.
UPDATE 2014-10-09: After several support tickets and chasing through Twitter I finally got the transfer code.. after two weeks! This clearly breaches the specified five calendar days to do the job.
Just a (hopefully) final note. If you do find that a registrar is being deliberately obstructive about the transfer (or they transferred a domain without your permission) you can make a complaint to ICANN here.