My cough was also productive. In December I posted:
"it could also be asthma. I developed a chronic cough about 5 years ago and jumped through all kinds of diagnostic hoops. It finally turned out to be asthma behaving in an unusual way. I tried a variety of inhaled steroids until I found one (Alvesco) with my allergist's help that made the cough go away."
There are a bunch of different breathing tests. The definitive one for asthma is called a "methacholine challenge." It's the only one that was able to make the diagnosis for me.
The process of figuring it out can be a big deal. My CT scan revealed a mass on my kidney that had to come out. It fortunately was benign, but the darn surgery wasn't.