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Fault detection is vital in ensuring AC microgrids' reliable and resilient operation. Its importance lies in swiftly identifying and isolating faults, preventing cascading failures, and enabling rapid power restoration. This paper proposes a strategy based on observers and residuals for detecting internal faults in grid-forming inverters with power-sharing coordination. The dynamics of the inverters are captured through a nonlinear state space model. The design of our observers and residuals considers $H_{-}/H_{\infty}$ conditions to ensure robustness against disturbances and responsiveness to faults. The proposed design is less restrictive than existing observer-based fault detection schemes by leveraging the properties of quadratic inner-boundedness and one-sided Lipschitz conditions. The internal faults considered in this paper include actuator faults, busbar faults, and inverter bridge faults, which are modeled using vector-matrix representations that modify the state space model of the inverters. One significant advantage of the proposed approach is its cost-effectiveness, as it does not require additional sensors. Experiments are conducted on an islanded AC microgrid with three inductive lines, four inductive loads, and four grid-forming inverters to validate the merits of the proposed fault detection strategy. The results demonstrate that our design outperforms existing methods in the field.