I get the same thing, and it definitely IS exercise induced asthma, regardless of whether you get the restricted-chest feeling or not. They respond to the same range of medications. I get the same thing happening as the previous poster. The aerosol Seretide helps my symptoms significantly.
The only time I get wheezy or feel restricted is if have a chest infection, but I always get the cough after a session where I've pushed to VO2 max, whether it's been running gymming or cycling. The only exception is on a surfboard due to the wet environment. But even then, sometimes, when the surf is big ... And it is always much worse when the air is cold, such as in the evenings.
What causes it is the large volumes of air you're passing through your windpipe. It dries out the airways and sensitises their walls.
The moisture and mucus lining your airways is a lot like the sticky oil in the wet air filters used in motor racing. It traps the ****, stopping it from enterng yor bloodstream, and small hairs called cilia sweep it back up so you can cough it up and keep your lungs clean.
When you dry out your airways by sucking and blowing huge amounts of air through them they respond by secreting more mucus to protect themselves, and send a message to your brain that they're irritated about being treated this way.
Hence the hacking cough.