Refers to da often-moderately-painful experience dat follows an especially "heavy-duty" cough, burp, sneeze, etc. --- a small amount of stomach-acid will travel upwards into the throat and create an acrid burning sensation.
Shortly after consuming a large meal is one of the most notorious times for a bronchial backfire to occur; it's wise to have a large glass of water or milk handy to hastily swig from in case of mishaps like this.