In this paper, we propose a scheme for generating genuine tripartite steering nonlocality in a cavity magnomechanical system composed of an yttrium iron garnet (YIG) sphere with a diameter of a few hundred micrometers inside a microwave cavity. In the system, the magnons, i.e., collective spin excitations in the sphere, are coupled to the cavity photons via magnetic-dipole interaction and at the same time coupled to phonons, the quanta of vibration of the sphere, by magnetostrictive interaction. We consider that the output field of the driven microwave cavity is subject to a time-continuous homodyne detection. We find that without the continuous measurement, only weak bipartite steering among the photons, magnons, and phonons can be obtained and the genuine tripartite steering is unachievable, although there exists weak genuine tripartite entanglement; when the continuous measurement is present, the bipartite steering is enhanced considerably and furthermore the genuine photon-magnon-phonon tripartite steering can be generated in the steady-state regime. It is shown that the generated tripartite steering is robust against thermal fluctuations for realistic parameters. Our scheme opens a promising route for exploring and exploiting macroscopic quantum effects in such a macroscopic quantum interface of photons, magnons, and phonons.