For a rod in compression, the max shear stress at a point is $\pm\sigma_x/2$, i.e. half the compressive stress applied at either end.
This value is obtained using the equation for $\sigma_\theta$, the longitudinal stress at any point in a material if it is subjected to a set of stresses (equivalently represented by normal and shear stresses), then rotated through an angle $\theta$, while these same stresses remain in place.
So, does this max shear stress in a rod in compression refer to when the rod is rotated by 45 degrees such that it experiences max shear stress under the same compressive load? Or is this a value that the rod experiences at all times as a result of weaknesses in the material along different axes? Because at an angle of 0 degrees, there wouldn't be any shear stress in a perfectly homogeneous rod under compression, would there?