This is a serious question, given my level of experience with this subject.

If my walls are constructed like this:
wood sliding
3/4" strapping
house wrap
1/2" plywood sheathing
2 layers of 2" foam
2"x6" studs
fiberglass insulation
vapor barrier
3/4" strapping
drywall

What type of fastener do I use to attach the all this to the 2"x6" studs?

Thanks for your patience. :)

I wouldn't put the foam in between the plywood sheathing and the studs. If the plywood is directly fastened to the studs, it plays a major role in the building's structure. With the foam (especially 4") in between, the frame would rack unless you want to go to the trouble to put bracing in. Putting the foam outside of the plywood, you would use long screws through the strapping and the foam into the plywood and studs. The strapping holds the foam to the wall. With that thickness of foam, I'd work out some way to support the bottom of the straps since the very long screws would have a lot of bending stress from the weight of the siding.

actually, using sufficient good screws, bending is not an issue. For instance, if you are strapping 16" or 24" OC vertically, 2 screws per stud will be fine

Use Pancake head timber screws 24" o.c. (eg: FastenMaster HeadLok, or similar) penetrating the stud by at least 1.5".

Screws with narrower heads it may split the furring and pull over time.

as jdebree said, you have the foam in the wrong place - this is the correct stackup. you use 7" tourque head screws - either GRK or Fastenal, or some of the TimberLoks also work - to fasten the foam. We tack the foam to hold it in place then fasten the strapping through the foam into the studs. Stagger the joints in the foam and tape the outside seams.

Also- this wall needs to breathe to the inside so lose the vapor barrier or use MemBrain; otherwise you're building a mold sandwich.

wood sliding
3/4" strapping
2 layers of 2" foam
house wrap
1/2" plywood sheathing
2"x6" studs
fiberglass insulation
vapor barrier
3/4" strapping
drywall

as jdebree said, you have the foam in the wrong place - this is the correct stackup. you use 7" tourque head screws - either GRK or Fastenal, or some of the TimberLoks also work - to fasten the foam. We tack the foam to hold it in place then fasten the strapping through the foam into the studs. Stagger the joints in the foam and tape the outside seams.

Also- this wall needs to breathe to the inside so lose the vapor barrier or use MemBrain; otherwise you're building a mold sandwich.

wood sliding
3/4" strapping
2 layers of 2" foam
house wrap
1/2" plywood sheathing
2"x6" studs
fiberglass insulation
vapor barrier
3/4" strapping
drywall

What Bob said about the interior vapor barrier. Inspectors in Canada don't always sign off on Certainteed MemBrain, but is a FAR better solution than poly sheeting or vapor-barrier latex paint when you have 4" of foam on the exterior.

This is a case of wording of the code interfering with good design.

Thanks for your answers. I understand that I placed the sheathing in the wrong place.

I don't understand the concern for moisture - with foam on the outside and VB on the inside, where is the moisture coming from?

Would it be better to place 2" foam on the outside of the studs and 2" of foam on the inside of the studs?

Also, do I still put house wrap over the plywood in any stackup?

Thanks, yet again.