This article introduces a novel complex sliding-mode control strategy for grid-forming inverters in islanded microgrids. This approach offers an alternative to the traditional hierarchical control strategies that typically rely on droop control, communication-based secondary controls, and virtual impedance methods. To develop the complex sliding-mode control, we propose a complex phasor model for grid-forming inverters, presenting an alternative to the conventional power flow model used in ac microgrids. This model enables the derivation of a linear complex dynamic model for grid-forming inverters connected to a microgrid, based on the apparent power differential equation, thereby simplifying the control law design process. The proposed control system features three key advantages: first, it ensures precise active and reactive power sharing across both resistive and inductive power lines, eliminating the need for virtual impedance; second, it exhibits high robustness against line impedance mismatches and external disturbances, including load changes and the connection and disconnection of inverters; and third, it eliminates the requirement for secondary frequency control. Experimental results validate the effectiveness of the proposed control strategy.