= Solution
The coupling $\frac c2\int\phi p^2$ is linear in $\phi$, but its coefficient $p^2(\mathbf r)$ is generally position dependent and dynamical. It therefore cannot be reduced to a constant times the conserved integral of $\phi$; it changes both the local chemical potential of the mixture and the tendency of the <polar order parameter> to order.
Interchanging the labels of the two species sends $\phi\mapsto-\phi$ while leaving the even terms in $\phi$ unchanged. It sends $c\mapsto-c$, so one may choose $c>0$ without loss of generality.
The isotropic disordered system is invariant under spatial inversion, which sends the <polar order parameter> $p\mapsto-p$ while leaving the scalar composition $\phi$ unchanged. A term $\phi p$ is odd under this symmetry and is forbidden; in more than one dimension it also fails to be a rotational scalar. Finally, if no higher even powers are retained, $b<0$ or $B<0$ makes the free energy unbounded below as the corresponding field grows. Thermodynamic stability therefore requires $b,B>0$.
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