The real and the symmetric nonnegative inverse eigenvalue problems are different

We show that there exist real numbers $\lambda _1,\lambda _2,\dotsc ,\lambda _n$ that occur as the eigenvalues of an entry-wise nonnegative $n$-by-$n$ matrix but do not occur as the eigenvalues of a symmetric nonnegative $n$-by-$n$ matrix. This solves a problem posed by Boyle and Handelman, Hershkowitz, and others. In the process, recent work by Boyle and Handelman that solves the nonnegative inverse eigenvalue problem by appending 0's to given spectral data is refined.