This is what I'm looking to do: Use two 12-foot 2x4's attached horizontally (near the top) to two 4x4 posts that are spaced 10 feet apart.

Would I need a third (middle) post? For looks, I want to avoid a third post. 2x6's for the horizontals is NOT an option.

To keep that likely warp, twist or bowing to a minimum, would it help ANY to screw two or three short sections of 4x4's between the 2x4s along the 10-foot space, sandwiching the two 2x4's together? Or would it then still warp, twist and bow together.

While my proposed ?structure?, LOL, will be otherwise and additionally attached to a larger existing garden trellis to stabilize it THAT part, please consider this proposed section a free-standing garden structure that will be seen mainly from the front, the front being the 12-foot length).

To get strong straight boards:
When you buy a board, look at the end grain. Get the boards with the most growth rings, spaced tightly together. Throw back ones that have the center of the tree, they will twist the most. Big surface knots are also to be avoided.