  Exotic General Massive Gravity is the next-to-simplest gravitational theory
fulfilling the so-called third-way consistency, the simplest being Minimal
Massive Gravity. We investigate the canonical structure of the first-order
formulation of Exotic General Massive Gravity. By using the Dirac Hamiltonian
formalism, we systematically discover the complete set of physical constraints,
including primary, secondary, and tertiary ones, and explicitly compute the
Poisson bracket algebra between them. In particular, we demonstrate that the
consistency condition for the tertiary constraints provides explicit
expressions which can be solved algebraically for the auxiliary fields $f$ and
$h$ in terms of the dreibein $e$. In this configuration, to confirm that the
theory is ghost-free, the whole set of constraints is classified into first and
second-class ones showing the existence of only two physical degrees of freedom
corresponding to one massive graviton. Furthermore, we identify the
transformation laws for all of the dynamical variables corresponding
essentially to gauge symmetries, generated by the first-class constraints.
Finally, by taking into account all the second-class constraints, we explicitly
compute the Dirac matrix together with the Dirac's brackets.
