= Solution
For nonzero real $g$, the continuous <internal symmetry of a classical field theory> is the <global U(1) symmetry>
$$
\phi\longmapsto e^{-i\alpha}\phi,\qquad
\Phi\longmapsto e^{-2i\alpha}\Phi,
\qquad \alpha\text{ constant}.
$$
The <kinetic terms> and mass terms are invariant separately, while the phases in $\phi^2\Phi^\dagger$ cancel. Independent phase rotations would instead require $2\alpha_\phi-\alpha_\Phi=0$ modulo $2\pi$, leaving one connected circle. Simultaneous <complex conjugation> is also a discrete symmetry, reversing the continuous charge; including it gives $U(1)\rtimes\mathbb Z_2\cong O(2)$ for generic parameters. At $g=0$ the continuous symmetry enlarges to $U(1)\times U(1)$.
The <Noether current> with particles assigned positive charges is
$$
\boxed{j^\mu=i\bigl(\phi^\dagger\partial^\mu\phi-(\partial^\mu\phi^\dagger)\phi\bigr)
+2i\bigl(\Phi^\dagger\partial^\mu\Phi-(\partial^\mu\Phi^\dagger)\Phi\bigr).}
$$
For example, put $A=\phi^2\Phi^\dagger$ and $B=\phi^{\dagger2}\Phi$. The <Euler-Lagrange equations> give $\partial_\mu j_\phi^\mu=ig(A-B)$ and $\partial_\mu j_\Phi^\mu=ig(B-A)$, with the factor $2$ already included in $j_\Phi^\mu$. Their sum vanishes. The <Noether charge> $Q=\int d^3x\,j^0$ is conserved and, after <normal ordering>, counts
$$
Q=N_\phi-N_{\bar\phi}+2(N_\Phi-N_{\bar\Phi}).
$$
Thus a $\Phi$ particle carries two units of the same <conserved charge> carried by a $\phi$ particle; their <antiparticles> carry the negatives. The interaction permits conversion of one $\Phi$ particle into two $\phi$ particles when allowed by <conservation of energy>, while preserving this weighted number.
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