Engine afterfire.

Mines done it since new. 17 years old bike now, dealer fitted stage 1. Still runs the same as it did then.
 
I had an intake backfire situation a few years ago. Blane found it and fixed it. One of the rubber boots going from the intake to the manifold was not seating correctly. Air was able to move back and forth at this point and I was getting some fire from it under the bear claw. If your intake has had some work, this may be an area to investigate.
 
How would i do that on a carburetor fitted bike? The dealer failed to fix it!

A good place to start is to verify the intake rubber boots and vacuum lines are OK and in place. Same with the exhaust header gaskets and packings ensuring everything is tight. Adjusting carb settings can be a lot more difficult than a fuel injected unit. Idle mixture screws, pilot jets, main jets, metering rod taper and height along with float height all work together to control the amount of fuel to air ratio your bike sees while running. The problem is learning what is in your carbs (needs to be taken apart) before moving forward with replacements/upgrades. This can involve purchasing replacement jets/metering rods/shims etc along with adjustments to the idle mixture screws. Also if you have access to a dyno or air fuel monitoring you can usually address the offending circuit in the carb but this takes time for every adjustment unlike EFI systems.

FWIW - I pulled my carbs on my America multiple times until I got rid of the decel popping and improved performance. Since my intake and exhausts were de-restricted I needed to increase fuel flow across the rev range to get rid of a mid range bog, lackluster performance, white spark plugs (too lean) and decel popping along with bluing pipes. My mixture screws were turned out, metering rods replaced with a more tapered set from a Thunderbird 900 and raised 2 mm to improve midrange. The main jets were increased several times till the right ones were found improving full throttle response. Carbs balanced. Plugs were checked multiple times till a nice light tan was realized. Drove the bike like this for many 1000s of kms till I sold it.
 
My plugs are a light grey /brown color after a run so i thought the general mixture was ok. It starts well too. It sounds like a mind blowing job and expensive but i have lived with it for 17 years now so i suppose it's gotta stay like it. The trouble is the Harley dealer fitted it all and then they gave up. Quite annoying really.
 
Hey, where abouts are you? Van Isle?