It is a fairly unusual problem but I for one have suffered from allergies for seventy years now and if I catch a winter cold, something in my area causes my allergies to flair up. After the cold has run it's course and if I haven't taken Zertex or something similar, my sinuses will drain and while I am asleep I aspirate the drainage. This irritates my bronchial tubes and I get a dry cough. While I have the cold I have a productive cough, post cold, I have a dry cough and it can last for several months.
When my nose starts running, allergies or cold, I take Zertex D. The rest of the year I take Zertex.