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Graphical abstract Abstract This work aims to investigate the changes in the shear and oscillation responses of an ethoxylated alkylamine emulsion with the addition of acetic acid and establishes their relationships with the emulsion phase inversion. The initial emulsion behaves as a non-Newtonian fluid exhibiting shear thinning with a viscoelastic gel structure (at rest). Upon the addition of acetic acid, the emulsion first transforms to a structureless (at rest) non-Newtonian fluid with lower viscosity and worse shear thinning compared with the initial state and then converts back to a non-Newtonian fluid with a higher gel strength, higher elasticity (at rest) and better shear thinning than the initial state as the phase inversion proceeds from an oil-continuous stage to a bicontinuous stage. This regained non-Newtonian character can be maintained throughout the whole bicontinuous stage and can even extend to the beginning of the water-continuous stage which is obtained from the bicontinuous stage by increasing the acetic acid content; however, it is ultimately replaced by a structureless (at rest) non-Newtonian fluid that exhibits lower viscosity and worse shear thinning compared with the initial state as the acetic acid content increases. The establishment of relationships among the interfacial moduli and deformation, geometrical factors (i.e., interfacial curvature, droplet shape and size and positional structure) and the liquid crystal phase behaviors help to elucidate the mechanisms that control the rheological transitions of emulsions. Based on these results, a novel water-washable viscoelastic emulsion drilling fluid is proposed. [ABSTRACT FROM AUTHOR]