The answer is trivially not, even in the topological sense. For example, you consider $\mathbb{CP}^2$, the toric fibration $f_1:\mathbb{CP}^2\rightarrow B_1$ is of course almost-toric, since only elliptic singularities are involved. Notice that in this case, $B_1$ is an triangle with its symplectic affine structure. In particular, $B_1$ is an integral affine manifold with corners. On the other hand, by the technique of "nodal trade" in the paper you referred to, $\mathbb{CP}^2$ admits another almost toric fiberation $f_2:\mathbb{CP}^2\rightarrow B_2$ whose base $B_2$ is diffeomorphic to the a disc with boundary. $f_2$ has 3 nodal fibers in the interior of $B_1$ whose topologies are given by $T^2$ with a cycle in $H^1(T^2,\mathbb{Z})$ collapsed to a point. In this case, $B_2$ is a singular integral affine manifold with boundary. In a similar way you can produce examples of almost-toric symplectic 4-manifolds which admits different almost-toric Lagrangian fibrations with different bases. For example, you can apply the nodal trade technique to an arbitrary del-Pezzo surface $X$. You can also consider non-compact examples, although this is not treated in the paper you referred to. For example, when $X=\mathcal{O}(-2)\rightarrow\mathbb{P}^1$. This is a non-compact toric variety, so there is a toric fibration whose base $B_1$ is a noncompact polytope $P\subset\mathbb{R}^2$ with two corners. On the other hand, you can pull the two corners into the interior of $P$ to get another almost-toric fibration $X\rightarrow B_2$. In this case, the new base $B_2$ is the half plane $\mathbb{R}\times\mathbb{R}_{\geq0}$, and there two nodal fibers. So $B_2$ is a non-compact singular affine manifold with boundary. This example is first considered by Mark Gross in his paper "Examples of special Lagrangan fibrations".

The answer is trivially no, even in the topological sense. For example, you consider $\mathbb{CP}^2$, the toric fibration $f_1:\mathbb{CP}^2\rightarrow B_1$ is of course almost-toric, since only elliptic singularities are involved. Notice that in this case, $B_1$ is an triangle with its symplectic affine structure. In particular, $B_1$ is an integral affine manifold with corners. On the other hand, by the technique of "nodal trade" in the paper you referred to, $\mathbb{CP}^2$ admits another almost toric fiberation $f_2:\mathbb{CP}^2\rightarrow B_2$ whose base $B_2$ is diffeomorphic to the a disc with boundary. $f_2$ has 3 nodal fibers in the interior of $B_1$ whose topologies are given by $T^2$ with a cycle in $H^1(T^2,\mathbb{Z})$ collapsed to a point. In this case, $B_2$ is a singular integral affine manifold with boundary. In a similar way you can produce examples of almost-toric symplectic 4-manifolds which admits different almost-toric Lagrangian fibrations with different bases. For example, you can apply the nodal trade technique to an arbitrary del-Pezzo surface $X$. You can also consider non-compact examples, although this is not treated in the paper you referred to. For example, when $X=\mathcal{O}(-2)\rightarrow\mathbb{P}^1$. This is a non-compact toric variety, so there is a toric fibration whose base $B_1$ is a noncompact polytope $P\subset\mathbb{R}^2$ with two corners. On the other hand, you can pull the two corners into the interior of $P$ to get another almost-toric fibration $X\rightarrow B_2$. In this case, the new base $B_2$ is the half plane $\mathbb{R}\times\mathbb{R}_{\geq0}$, and there are two nodal fibers. So $B_2$ is a non-compact singular affine manifold with boundary. This example is first considered by Mark Gross in his paper "Examples of special Lagrangan fibrations".