  The generation and manipulation of strong entanglement and
Einstein-Podolsky-Rosen (EPR) steering in macroscopic systems are outstanding
challenges in modern physics. Especially, the observation of asymmetric EPR
steering is important for both its fundamental role in interpreting the nature
of quantum mechanics and its application as resource for the tasks where the
levels of trust at different parties are highly asymmetric. Here, we study the
entanglement and EPR steering between two macroscopic magnons in a hybrid
ferrimagnet-light system. In the absence of light, the two types of magnons on
the two sublattices can be entangled, but no quantum steering occurs when they
are damped with the same rates. In the presence of the cavity field, the
entanglement can be significantly enhanced, and strong two-way asymmetric
quantum steering appears between two magnons with equal dispassion. This is
very different from the conventional protocols to produce asymmetric steering
by imposing additional unbalanced losses or noises on the two parties at the
cost of reducing steerability. The essential physics is well understood by the
unbalanced population of acoustic and optical magnons under the cooling effect
of cavity photons. Our finding may provide a novel platform to manipulate the
quantum steering and the detection of bi-party steering provides a knob to
probe the magnetic damping on each sublattice of a magnet.
