Hello Chris
As far as I know, there is no standard way to use T2D for a non-newtonian fluids. Debris flows is one of the main development axis on which Telemac team will focus on the near future. For mud, I don't know if there is a valid way to handle it using Telemac+sisyphe. I leave my colleagues complement the answer.
A very basic idea to start with is the implementation of your (visco-elastic or visco...) law and introduce it as a variable viscosity (such as for turbulence models for instance). This is really a very simplistic way, a real true implementation needs, for sure, more time and efforts.
With my best regards
Riadh ATA