Strangely enough my wife has exactly what you describe and has had so for about 5 years. Her problem is a Hiatus hernia, Barrets Syndrome and reflux.
I have asked about it on AB before as she propably had 6 or 7 chest infections in a year, each one needed Penecilin to clear it up.
She stopped taking Omeprazole and the chest infection stopped, although the coughing persists ... as soon as she starts taking it again, another infection arrives.
It really is a viscious circle between Omeprazole, infections and lack of sleep through constant coughing.