The eventually positive inverse eigenvalue problem (EPIEP) asks when a list $\sigma=(\rho,\lambda_2,\ldots,\lambda_n)$ of $n$ complex numbers is the spectrum of a real $n\times n$ eventually positive matrix. In this paper, a constructive method is presented to solve the EPIEP. Using this method, it is easy to find the realizing matrix, i.e., a real eventually positive matrix with the spectrum $\sigma$. And a complete answer to the EPIEP prescribed the left and the right Perron eigenvectors of the realizing matrix is also given.