While not tartan, a LOOOONG time ago, Structure (Express - Men) had a series of shirts that looked like tattersall, windowpane, and madras that were of a very lightly woven material. It was almost mesh, but looked like regular fabric. The individual threads were large, but the weave allowed air to go right through it.

I still have four of them, but haven't seen their ilk for sale in nearly 15 years. I wish I could find some more of them, after being dragged from beach vacation to beach vacation for so many years they are starting to look their age.