  In this article, we consider the following class of stochastic partial
differential equations (SPDE): \begin{equation*}
\left\{\begin{aligned}\mathrm{d}
\mathbf{X}(t)&=\mathrm{A}(t,\mathbf{X}(t))\mathrm{d}
t+\mathrm{B}(t,\mathbf{X}(t))\mathrm{d}\mathrm{W}(t)+\int_{\mathrm{Z}}\gamma(t,\mathbf{X}(t-),z)\widetilde{\pi}(\mathrm{d}
t,\mathrm{d} z),\; t\in[0,T],\\ \mathbf{X}(0)&=\boldsymbol{x} \in
\mathbb{H},\end{aligned} \right.\end{equation*} with fully locally monotone
coefficients in a Gelfand triplet $\mathbb{V}\subset
\mathbb{H}\subset\mathbb{V}^*$, where the mappings \begin{align*}
\mathrm{A}:[0,T]\times \mathbb{V}\to\mathbb{V}^*,\quad \mathrm{B}:[0,T]\times
\mathbb{V}\to\mathrm{L}_2(\mathbb{U},\mathbb{H}), \quad
\gamma:[0,T]\times\mathbb{V}\times\mathrm{Z}\to\mathbb{H}, \end{align*} are
measurable, $\mathrm{L}_2(\mathbb{U},\mathbb{H})$ is the space of all
Hilbert-Schmidt operators from $\mathbb{U}\to\mathbb{H}$, $\mathrm{W}$ is a
$\mathbb{U}$-cylindrical Wiener process and $\widetilde{\pi}$ is a compensated
time homogeneous Poisson random measure. Such kind of SPDE cover a large class
of quasilinear SPDE and a good number of fluid dynamic models. Under certain
generic assumptions of $\mathrm{A},\mathrm{B}$ and $\gamma$, using the
classical Faedo-Galekin technique, a compactness method and a version of
Skorokhod's representation theorem, we prove the existence of a
\emph{probabilistic weak solution} as well as \emph{pathwise uniqueness of
solution}. We use the classical Yamada-Watanabe theorem to obtain the existence
of a \emph{unique probabilistic strong solution}. Finally, we allow both
diffusion coefficient $\mathrm{B}(t,\cdot)$ and jump noise coefficient
$\gamma(t,\cdot,z)$ to depend on both $\mathbb{H}$-norm and $\mathbb{V}$-norm,
which implies that both the coefficients could also depend on the gradient of
solution. We establish the global solvability results.
