In the regime of strong light-matter coupling, polariton modes are formed that are hybrid light-matter excitations sharing properties of both, an electrodynamic cavity mode and a matter mode. In the recent decade, magnon-polaritons have been intensively researched using ferromagnetic materials in the microwave range, with potential applications for quantum technology and sensors. Exploring antiferromagnets raises magnon-polariton frequencies into the terahertz (THz) range. In this range, many dielectric excitations like phonons, vibrational modes of molecules, plasmons in two-dimensional electron gases, etc, are characterized by higher light-matter coupling rates than those of magnetic excitations because of their high dipole moments. Here, we report on the cavity-mediated coupling of phonons and magnons, excitations of different natures. We used magnon in nickel oxide (NiO) owing to its low damping, and controllable frequency in the range of 0.7-1.0 THz using temperatures above room temperature. We report on the coupling of its magnon mode to a phonon mode at 0.92 THz in CuB₂O₄ ceramics. Our experimental setup consists of parallel-plane slabs of both materials, placed next to each other at a well-controlled gap, forming a tunable Fabry-Pérot-type cavity. At the condition of matching frequencies of a cavity mode, magnon, and phonon, we observed tripartite phonon-magnon-polariton modes. They share properties of a cavity mode, the magnon that has magnetic dipole moment only in its pure form, and the phonon that normally has electric dipole moment only. This hybridization is possible without directly interfacing the two materials, at distances up to a few mm long