  We construct the effective field theory for time-reversal symmetry breaking
multi-Weyl semimetals (mWSMs), composed of a single pair of Weyl nodes of
(anti-)monopole charge $n$, with $n=1,2,3$ in crystalline environment. From
both the continuum and lattice models, we show that a mWSM with $n>1$ can be
constructed by placing $n$ flavors of linearly dispersing simple Weyl fermions
(with $n=1$) in a bath of an $SU(2)$ non-Abelian static background gauge field.
Such an $SU(2)$ field preserves certain crystalline symmetry (four-fold
rotational or $C_4$ in our construction), but breaks the Lorentz symmetry,
resulting in nonlinear band spectra (namely, $E \sim (p^2_x + p^2_y)^{n/2}$,
but $E \sim |p_z|$, for example, where momenta ${\bf p}$ is measured from the
Weyl nodes). Consequently, the effective field theory displays $U(1) \times
SU(2)$ non-Abelian anomaly, yielding anomalous Hall effect, its non-Abelian
generalization, and various chiral conductivities. The anomalous violation of
conservation laws is determined by the monopole charge $n$ and a specific
algebraic property of the $SU(2)$ Lie group, which we further substantiate by
numerically computing the regular and "isospin" densities from the lattice
models of mWSMs. These predictions are also supported from a strongly coupled
(holographic) description of mWSMs. Altogether our findings unify the field
theoretic descriptions of mWSMs of arbitrary monopole charge $n$ (featuring $n$
copies of the Fermi arc surface states), predict signatures of non-Abelian
anomaly in table-top experiments, and pave the route to explore anomaly
structures for multi-fold fermions, transforming under arbitrary half-integer
or integer spin representations.
