Lagrangian submanifolds from tropical hypersurfaces

We prove that a smooth tropical hypersurface in $mathbbR^3$ can be lifted to a smooth embedded Lagrangian submanifold in $(mathbbC^*)^3$. This completes the proof of the result announced in the article "Lagrangian pairs pants" arXiv:1802.02993. The idea of the proof is to use Lagrangian pairs of pants as the main building blocks.