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Active acoustic metamaterials incorporate electric circuit elements that input energy into an otherwise passive medium to aptly modulate the effective material properties. Here, we propose an active acoustic metamaterial with Willis coupling to drastically extend the tunability of the effective density and bulk modulus with the accessible parameter range enlarged by at least two orders of magnitude compared to that of a non-Willis metamaterial. Traditional active metamaterial designs are based on local resonances without considering the Willis coupling that limit their accessible effective material parameter range. Our design adopts a unit cell structure with two sensor-transducer pairs coupling the acoustic response on both sides of the metamaterial by detecting incident waves and driving active signals asymmetrically superimposed onto the passive response of the material. The Willis coupling results from feedback control circuits with unequal gains. These asymmetric feedback control circuits use Willis coupling to expand the accessible range of the effective density and bulk modulus of the metamaterial. The extreme effective material parameters realizable by the metamaterials will remarkably broaden their applications in biomedical imaging, noise control, and transformation acoustics-based cloaking.