Existing methods to find the eigenvalue spectrum (or a reasonable approximation to it) of square matrices can be extended to Stochastic Matrices (SM). The matter is more delicate for the Inverse Eigenvalue Problem (IEP), which consists in the reconstruction of a matrix from a given eigenvalue spectrum. In this work, we present a simple method to solve a real-valued IEP for SM by constructing step-by-step the solution matrix through an elementary Markov state disaggregation method named state splitting, and based on a matrix operator. After showing some results on how the splitting operator influences the steady-state distribution of the Markov chain associated with the SM, we demonstrate that the state splitting operator has a fundamental property: when applied to a SM A of size n-by-n, it yields a SM of size (n+1)-by-(n+1), whose eigenvalue spectrum is equal to that of A, plus an additional eigenvalue belonging to a bounded interval. We use a constructive method to prove that for any spectrum made of real and positive eigenvalues, one can build up an infinite number of SM sharing this spectrum. Finally, we present a new sufficient condition to test if a given set of real values can be the spectrum of a SM constructed by the proposed method.