In string compactifications on Calabi-Yau threefold hypersurfaces in toric varieties, a choice of fine, regular, star triangulation of a 4D reflexive polytope determines one geometric phase. The full Kähler moduli space relevant for many physical questions is instead organised by the extended Kähler cone: the union of Kähler cones of Calabi-Yau threefolds in a birational equivalence class, glued along walls corresponding to flop transitions. This structure is natural for both algebraic geometers and string theorists, but it is rarely built directly into numerical searches for string vacua. I will introduce FanRoots, a framework for moduli-dependent potentials and residual objectives while navigating the phases of the extended Kähler cone. The method follows optimisation trajectories through the fan, detects wall crossings, and updates the geometric data associated with the active phase. I will explain the geometric motivation and discuss two applications: Kähler moduli stabilisation across extended Kähler cones, and complex structure moduli stabilisation in Type IIB flux compactifications via the mirror description.