I just bought a JTV-59 korea model 2 days ago and am having this exact problem.
In most tunings like drop dflat and open G, open D, open A, the top three strings sound like a 12 string guitar but not exactly, it sounds like the original sound is also leaking through. This is most apparent when using clean models like acoustic or reso, making it unplayable.
I would like to know if this is a problem that can be fixed by software update or is it a hardware problem and I have to replace it?
Im going to try and install the latest update tonite, if mine already isnt the latest one and post back the results.
BTW, Im connecting the Guitar to my POD XT Live via the Variax Input. Output is from the phones to my Studio monitors. All amps, cabs, mods and effects are turned off on the pod xt live.