Research in auto-sensing materials is crucial for smart structural health monitoring of civil engineering structures through digital twins~[1]. This paradigm requires mapping the strain field through a relation between the electrical field and the strain field that must hold for both low strain levels and material rupture. This motivates the development of carbon nanotube-based (CNT) composites that can bring auto-sensing capabilities due to the interactions between the nanofillers by the so-called piezoresistive effect.  This work proposes a mathematical framework to simulate deformation and fracture of CNT-based composites in an electro-mechanical setting. First, the electro-mechanical properties are estimated by employing micromechanical principles, which serve as an input in the finite element model. Then, an electro-deformation-fracture problem is numerically solved, making use of a phase-field fracture formulation~[2, 3]. This variational approach estimates the crack propagation in arbitrary geometries based on Griffith's fracture theory. Finally, the electrical conductivity of the composite is coupled with the phase-field variables, degrading itself due to the loss of mechanical stiffness. The case studies addressed showcase the capabilities of the model in capturing the interplay between fracture and electromechanical material behaviour in CNT-based composites.