In the paper "Infinently many monotone Lagrangian tori in del Pezzo surfaces" by Renato Vianna, the author construct an infinite ammount of non symplectomorphic monotone Lagrangian tori in del Pezzo surfaces.

From what I understand, the basic idea is the following. Let's focus on the case of $\mathbb{C}\mathbb{P}^2$ so the Delzant polytope is a triangle and the central fiber is known to be monotone. Then we apply nodal trades to the vertices, and let them slide trough this monotone fiber. Then we want to visualize the fibration in another way so that the new created fiber does not lie on the eigenline, and we apply a mutation so that we are able to visualize the desired fiber. 

What I fail to understand is why is the fiber that will replace the previous monotone fiber be monotone ? Is this a straightforward thing to see or is there some depth to it ?
Any insight is appreciated.