The authors study the induced spherical ensemble of non-Hermitian matrices with real quaternion entries. They use the induced spherical ensemble as those \(N\times N\) matrices \(G\) that are defined by the matrix probability to define the ensemble by the matrix probability distribution function that is a density function proportional to \(\frac{\det(GG^+)^{2L}}{\det(1_N+GG^+)^{2(n+N+L)}}\). Such matrices can also be constructed via a procedure called ‘inducing’, using a product of a Wishart matrix (with parameters \(n\), \(N\)) and a rectangular Ginibre matrix of size \((N+L)\times N\). Using functional differentiation of a generalized partition function, they make use of skew-orthogonal polynomials to find expressions for the eigenvalue \(m\)-point correlation functions, and in particular the eigenvalue density (when \(m=1\)). After a stereographic projection, the density is uniform on a spherical annulus, except for a depletion of eigenvalues on a great circle corresponding to the real axis (as expected for a real quaternion ensemble). The authors also form a conjecture for the behavior of the density near the real line based on analogous results in the \(\beta=1\) and \(\beta =2\) ensembles.