Our expert says:
Wow, difficult question. Like you, I'd have though the smoking was the cause. It may still be linked, of course, perhaps there is a longer-lasting effect than we both would have suspected.
The other possibility is that you have some kind of reflex bronchoconstriction, which sets in after exercise, much like asthma, except yours is not a shortness of breath but rather an irritation of the airways that causes you to cough.
If you have had tests on the lungs specifically, then the good news is that it sounds like there's no major issue there. The history of asthma is also likely significant, suggesting what I mentioned earlier as a cause. Have you tried using a bronchodilator pump before and immediately after exercise, or carrying one with you while you run? That may clear it up right away, which kind of gives you an answer, if not a diagnosis/cause? For example, there is exercise induced asthma, and perhaps in a mild form, it's causing enough constriction to irritate your airways and to cough.
So I'm afraid my only suggestion is to try the bronchodilator before and after training. And possibly to have it tested, but hopefully the pump works.
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