Geologic discontinuities usually exist in subsurface permeable formations, where multiple reservoir regions with distinct properties are separated by linear leaky faults. This kind of heterogeneous reservoir is usually called a linear composite reservoir. Although many analytical/semi-analytical linear composite models have been established to investigate the pressure behavior for linear composite reservoirs, almost all of these models were aimed at vertical wells without hydraulic fracturing and there are few analytical/semi-analytical models of fractured vertical wells in linear composite reservoirs. This paper first derives the Laplace-space point source solution for anisotropic linear composite systems separated by a partially communicating fault. Then, superposition principle and fracture discrete scheme are employed to acquire the semi-analytical solution for finite-conductivity fractured vertical (FCFV) wells in anisotropic linear composite reservoirs with a fault. The proposed solution is validated against numerical solutions under different reservoir scenarios. The characteristic of the pressure behavior for an FCFV well in anisotropic linear composite reservoirs with a fault is discussed in detail. The proposed model can be employed to obtain accurate pressure response with high computational efficiency. It is a good start to further develop analytical/semi-analytical models for other complex well types in an anisotropic linear composite reservoir with a fault.