A rapid and anisotropic modification of the Fermi-surface shape can be associated with abrupt changes in crystalline lattice geometry or in the magnetic state of a material. In this study we show that such an electronic topological transition is at the basis of the formation of an unusual pressure-induced tetragonal ferromagnetic phase in Fe$_{1.08}$Te. Around 2 GPa, the orthorhombic and incommensurate antiferromagnetic ground-state of Fe$_{1.08}$Te is transformed upon increasing pressure into a tetragonal ferromagnetic state via a conventional first-order transition. On the other hand, an isostructural transition takes place from the paramagnetic high-temperature state into the ferromagnetic phase as a rare case of a `type 0 transformation with anisotropic properties. Electronic-structure calculations in combination with electrical resistivity, magnetization, and x-ray diffraction experiments show that the electronic system of Fe$_{1.08}$Te is instable with respect to profound topological transitions that can drive fundamental changes of the lattice anisotropy and the associated magnetic order.
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