Let $(M,\omega)$ be a closed, symplectic four-manifold admitting an almost toric fibration, in the sense of Symington and Leung (e.g. http://arxiv.org/pdf/math/0210033.pdf). That is, there is a Lagrangian torus fibration $$\pi\,:\,(M,\omega)\rightarrow B$$ such that the fibers attain at worst nodal or elliptic singularities. Then, $B$ has the natural structure of an integral affine-linear surface and is the natural generalization of the moment polygon. As far as I can tell, Symington proves that given an appropriate base $B$, there is a unique symplectic manifold $(M,\omega)$ admitting a unique almost toric fibration over $B$.
My question is the following; it may be very stupid, since I don't know too much symplectic geometry. Let $(M,\omega)$ be a symplectic four-manifold that admits some almost toric fibration. Is there a unique base $B$ over which $(M,\omega)$ admits an almost toric fibration? (Emphasis on uniqueness of the base!)
In case this turns out to be false in general, I'm specifically interested in the case where $M$ is a rational algebraic surface.