Can one construct a Geometric Algebra to track a polarization state and modifications to said state with more convenience than existing formalisms like formalisms like Stokes vectors and Müller matrices or coherency matrices?

The Stokes formalism has the hassle that one has to represent the state along a path(a Stokes Vector) and modifications(Müller matrices) to said state separately, On the other hand coherency matrices treat the state and modifications both with 2x2 matrices with complex valued elements where the hassle instead comes from the complex numbers and greater difficulty extracting radiance of the path (i.e. taking the trace of the matrix).

Furthermore in both formalisms one has to align reference frames between incoming and outgoing for the state modifications to be physically valid, usually done with rotation matrices.

As has been explained in the GA SIGGRAPH 2019 talk, matrices are considered a less elegant way than Geometric Product to do rotations and translations.

For more details on polarization in renderers see:

Polarised light in computer graphics,

Bidirectional Rendering of Polarized Light Transport,

How to write a polarisation ray tracer

Edit: Came across a 3D Coherency Matrix Formalism which is in global coordinates (i.e. no reference frame rotations)

Three-dimensional polarization ray tracing calculus for partially polarized light