Controlling dynamical mode coupling in magnetic nano-oscillators is essential to improving their microwave spectral properties for rf applications and neuromorphic computing. Progress here is held back by the lack of a suitable platform that provides controlled coupling. This study experimentally demonstrates control of dynamical mode coupling in spin Hall nano-oscillators with perpendicular magnetic anisotropy, by means of temperature, excitation current, and magnetic field. It is established that mode coupling in this system is dominated by thermal magnon-mediated scattering, suggesting fresh approaches to engineering device properties suitable for the desired applications.