Transformer saturation is a common issue in megawatt converter-based grid emulators (GEs) when emulating grid faults. This problem necessitates the use of anti-saturation control (ASC) with GEs. However, conventional ASC methods tend to distort the emulated grid voltage or even interact with the voltage control (VC) of GEs, causing instability of the system. This paper, thus, proposes an adaptive ASC method and superimposes its output command to both modulation and VC references, which not only alleviate transformer saturation with the minimized voltage distortion, but mitigate its adverse interaction with the VC of converter-based GEs. Experimental results confirm the effectiveness of the adaptive ASC.