Tim,
Spitting and coughing is indication of weak mixture.
It's useful to establish whereabouts the spitting is occuring in relation to throttle position/load/rpm, to establish what you need to adjust to solve the problem.
The carbs essentially have a 2-stage jet system - idle jets control the mixture at idle and low-load low throttle openings. The main jets (in conjunction with Emulsion Tubes) then take over to control mixture in the midrange to high load conditions.
So if the popping is happening at idle or low-load conditions (which is the most usual) then the idle jet size needs increasing for example.
The popping at the exhaust on overrun is often due to ignition timing being too retarded.