Even if you have a non-linear system to solve, you should still be able to solve the system using numerical techniques. Is this something you have tried?
So I have a series of experimentally determined 2x2 matrices (y0, y1, y2...) and I know each of them is a product of two 2x2 matrices (a, b, and c). Ultimately I have three unknown matrices and six equations as follows:
[y0] = [a][b]
[y1] = [b][a]
[y2] = [a][c]
[y3] = [c][a]
[y4] = [b][c]
[y5] = [c][b]
These matrices will not always be diagonalizable but sometimes they are (in which case its not a problem).
I have found a solution to get an equation with one of the unknown matrices squared (ex: [y0][y5]^-1[y3] = [a][a]) but I'm not sure if the square root will always work.
If you take the numbers out of the matrices it seems like there should be 24 eqautions (6*4) but only 12 unknowns (3*4) but that may be not true because they aren't linear.