  We construct Poisson bracket relations between the operators which generate
the chiral ring of the Coulomb branch of certain $3d$ $\mathcal{N}=4$ quiver
gauge theories. In the case where the Coulomb branch is a free space, $ADE$
Klein singularity, or the minimal $A_2$ nilpotent orbit, we explicitly compute
the Poisson brackets between the generators using either inherited properties
of the abstract Coulomb branch variety, or the expected charges of these
operators under the global symmetry (known through use of the monopole
formula). We also conjecture Poisson brackets for Higgs branches that originate
from $6d$ theories with tensionless strings or $5d$ theories with massless
instantons for which the HWG is known, based on representation theoretic and
operator content constraints known from the study of their magnetic quiver.
