I also had the same symptoms and the reason was the primary TPS. Replaced it and got the voltage right -> no more backfire or hiccups. With the broken TPS the bike was running really bad.
 
I am getting desperate and tired of the Rocket 3 Touring running badly. I am hoping I can get some help.
Here is my problem. The bike will pull hard and really go when I get on the throttle. It idles fine , without a hiccup. But when I am just cruising along , I get a backfire threw the intake or by the air filters, it can do this cough 2 or 3 times in a row and run good for a bit and then cough again. It seems worse when it warms up or it is hot outside [air temperature].
The bike is an 2008 Touring and I dont believe it has the beaver tail relay.
Any and all help would be appreciated. Keep your head up and the rubber side down.

We just worked on a 2005 R3 & the problem we had was that it was sucking in air from the mounting boot on the intake stakes against the block. Since we didn't have spares we changed bolts and install it with ( Grey Gasket Make Grey ) to make sure it seal perfectly and this did the trick.Make sure you use minimal amount so that it does not go into the intake passages. This did the trip.
 
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