Haze on inside of windshield

I have recently inherited by father-in-law's 2000 Intrepid, which he bought new. I would tease him about not being able to see out the windshield due to a haze on the inside of the glass -- it looks a lot like smoke residue. Now that I have it, I realize that I cannot get rid of the haze; unless I really really work at it, window cleaner just seems to smear it around. Even if I do get the glass clean, it is back within 24 hours. I would assume that after 4 years, the dashboard, defroster ducts, etc. would have finished outgassing, so I can't figure out what this stuff is, how to clean it off, or how to get rid of it permanently.
Anyone else have any experience with this problem, or any suggestions on how to deal with it?

Thanks!
Mark

Sounds like your heater core is starting to go. In my other car, I have a 2004 Dakota and within months of buying it, I noticed a haze on my windshield. At the same time, it was leaking coolant on my feet and getting all over my rubber floor mats. You can clean it off all you want but it sounds like your heater core is going bad.