It is well known that the perceived texture and consistency of liquid foods are strong drivers of consumer preference, yet quantification of these parameters is made complicated by the absence of a concise mathematical framework. In this paper, we demonstrate that fractional rheological models, including the fractional Maxwell model (FMM) and the fractional Jeffreys model (FJM), are potential candidates to fill this void as a result of their ability to succinctly and accurately predict the linear and nonlinear viscoelastic response of a range of liquid food solutions. These include a benchmark fluid, the dysphagia product Resource(R) Thicken Up Clear, various plant extracts whose constituent polysaccharides have been reported to impart significant viscoelasticity, and human whole saliva. These fractional constitutive models quantitatively describe both the linear viscoelasticity of all of the liquid foods as well as the shear thinning of their steady shear viscosity (through application of the Cox-Merz rule), and outperform conventional multi-mode Maxwell models with up to 50 physical elements in terms of the goodness of fit to experimental data. Further, by accurately capturing the shear viscosity of the various liquid food solutions at the shear rate of γ = 50 s^-1 (widely deemed relevant for oral evaluation of liquid texture), we show that two of the constitutive parameters of the fractional Maxwell model can be used to construct a state diagram that succinctly characterizes both the viscous and elastic properties of the different fluids. This characterization facilitates the assignment of quantitative values to largely heuristic food textural terms, which may improve the design of future liquid foods of specific desired consistencies or properties.