A triple mixed-spin Ising system defined on the Bethe lattice is numerically investigated by means of exact recursion relations (ERRs) calculations. The lattice is constituted by three types of magnetic atoms A, B, C with spins 1/2, 1, 3/2 respectively arranged in the form ABCABC. The effects of bilinear exchange and crystal-field interactions as well as those of thermal fluctuations on the order parameters and phase diagrams are thoroughly studied and specified. First-order transitions and tricritical points are present for the coordination number q = 4 whereas at q = 3 they are absent. Global compensation phenomena are absent for the magnetic system. Instead, it is shown that it can only occur between the sublattice magnetizations B and C of the system. Several novel kinds of reentrance of the phase boundaries while varying the values of model parameters have been reported.