Invariance of spin coefficients

I have a question about how spin coefficients (Newman Penrose formalism) transform.

I know that if we perform a tetrad rotation, say of Class III:

$(l,n,m,\overline{m})\mapsto \left(\frac{1}{A}l, An, e^{i\theta}m, e^{-i\theta}\overline{m}\right)$

then the spin coefficients transform also, as functions of $A,\theta$ and their derivatives.

Now, my question is: assume I'm working in a spacetime, and want to do a coordinate transformation (not a tetrad rescaling!) Would it make sense to say that the spin coefficients should be invariant under any coordinate transformation? Is it always true/not true?

Thank you for any hints.

What changing coordinates does is that it changes the coordinate representation of the tetrad vectors (in terms of the coordinate vector fields $\partial^\alpha$), but the geometry (which is captured by the relationships between the spin coefficients) remains the same.