Surface tension force plays a central role in several two-phase flow applications, including rising bubbles, droplets in shear flows and topology changes (e.g., breakup and coalescence). Due to its interfacial nature, an accurate discretisation of such a force term is one of the most challenging aspects of numerical simulations of interface-resolved two-phase flows. Volumetric formulations, such as the Continuous Surface Force (CSF) method, have become the standard approach to treat surface tension, owing to the well-balancing of pressure gradients and surface tension. Such approaches come with the drawback of discretising the interfacial delta function, which necessarily results in a smoothing of interfacial forces and, therefore, in the loss of a truly sharp representation of the interface. The CSF method is known to be not conservative and additional challenges arise when the surface tension coefficient $(\sigma)$ is variable, due to an extra term (the surface gradient of $\sigma$) that is non-trivial to discretise. An alternative to the volumetric treatment of surface tension is the integral formulation, first introduced by Popinet1999 in a two-dimensional front-tracking scheme, which consists of a geometric computation of the surface tension force at the intersections between the interface and the faces of each computational cell. This method is inherently conservative and readily applicable to variable surface tension problems. The two main limiting factors are the difficulties (especially for three-dimensional flows) in the computation of the intersections between the interface and the cells, and the lack of the well-balancing property, which cannot be guaranteed. To the best of the authors' knowledge, only two works have recently proposed a (two-dimensional) implementation of the integral surface tension scheme [Abu-Al-Saud2018, Saini2025], for the level-set and volume-of-fluid methods, respectively. No attempts to design a three-dimensional scheme have been made. In the present work, we propose an implementation of the integral surface tension method in our state-of-the-art three-dimensional front-tracking framework [Gorges2025]. The scheme is tested for a range of two-phase configurations, including static and translating droplets, as well as flows with variable surface tension. Results are compared against the CSF formulation and analytical results, and the balancing between pressure gradients and surface tension is discussed.