When I first learned General Relativity, the tetrad formalism was introduced with near simultaneity. I was immediately taught that, to utilize spinors in any way, I had to formulate a local orthonormal frame. I learned to live with that just fine.
Lately, I've been coming across references and methods for utilizing spinors in curved spaces (and oblique coordinates to put it crudely). First I was reading a paper and found references to a paper:
Ogievetskii, V.I. & Polubarinov, I.V.. (1965). ON SPINORS IN GRAVITATION THEORY. Zh. Eksperim. i Teor. Fiz.. Vol: 48.
Where they talk about the no-go theorem only applying to spinors when a linear connection is used. There are quite a few more recent papers talking about utilizing spinors in curved spacetime without tetrads because they're using a nonlinear connection. here and here for instance.
I now totally get that you can write a tetrad in Minkowski space in terms of a complex Dirac spinor (I'm still learning the Newman-Penrose formalism, but I think of it kind of as a null vector field); however, the idea of utilizing them in full curved space? I've seen nonlinear connections used in other contexts so it seems kosher to me. Can someone straighten me out on this? Do we really need tetrads?