Any trike *can* be made to tour but what trike will work best. If you are talking fully self supported, you are going to be carrying some serious gear. The terrain you are talking will have some serious climbs... and decents. Comfort level requirements too have to be considered.
Panniers or trailers? Gearing? Suspension? Chromoly or Aluminum? Which tires and rims?
...and how is the condition of the engine?
I agree that the ICE TRICE Q or QNT or better yet the ICE-T would make a nice touring trike... and with a load, the optional drag brake is a really nice feature to have going down long decents. The tooth rattling chipseal will make suspension pay for itself, as well as large wide tires at low pressure. Also consider raising gear range with various drive options. Panniers don't get flats but on a suspended trike, setting things up takes a little attention but not undoable.
But also consider your budget... as a nicely packed out Trice can run you into the near $4K USD/CAN
Even a Catrike can get you up in the $3K quickly... and no suspension; stiff frame [more efficient]; no drag brake... but many options for the *right* trike for you. Still may need some gearing tweaks for the hills, though.
...And if budget is no problem, throw Greenspeed and HPVelotechnik Scorpion into the mix.
I'd say that if minimalist is your style... go Catrike... but if you are a gadget horder, TRICE. If you like the technology of Europe... HP Velotechnik Scorpion or Scorpion FX... and finally... BeaM'R, Mercedies or even Volvo man... go Greenspeed.
These are my observation over the past few years and everyone elses milage may vary... but you asked