Calculating smallest eigenvalues by real part using Arnoldi method

Wolfram Tech Support has confirmed that this is a bug and will work on fixing it.

Probably the "Shift" option is not being properly set for the given "Criteria". Since we know that the largest eigenvalue by magnitude is 2.35168, we can assume that the smallest possible eigenvalue by real part will have a real part no smaller than -2.35168. So, using this as an approximate shift gives:

Eigenvalues[mat, -2, Method -> {"Arnoldi", "Criteria"->"RealPart", "Shift"->-2.5}]

{-2.35168, -0.48013}

as desired.