In my limited experience a broken downpipe will cause plenty of coughing & backfiring even on an otherwise properly-tuned motor. I would theorize that it's a combination of a lack of backpressure and the introduction of extra oxygen (from the break in the pipe) so that under deceleration, when the mixture's not usually a perfect 14.7:1 stoichiometric ratio anyhow (especially if you've snapped the throttle closed causing high vacuum while a fuel charge is still on its way down the intake runners), those superheated exhaust gas pulses still contain some unburned fuel which instead of being cooled and dissipated through the exhaust are reigniting when the oxygenated air is suddenly introduced a few inches downstream. Again, it's just a theory (my attempt to prove I could have been an engineer, I guess) and as George pointed out there's no substitute for proper baseline settings. Just bear in mind that drastically changing backpressure does change the flow dynamics of the the engine. Case in point, a 16v VW I drove a while back with headers & bloody big wide-open exhaust, but no other mods. Loud, yes. Did it work any better? Not at all... Seemed worse, in fact. My old 8v Golf outran it hands-down. And have you noticed that most of the little rice rockets running around with big exhausts pop & backfire under deceleration? Of course, they don't really count, anyway...
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Chris, Dartmouth NS Canada 70 M-B 280SE, 83 245DL, 84 244 turbo, 90 780 turbo, 92 VW Golf, 90 740 Rex/Regina