We show how to obtain all the models of the continuous description of membranes by constructing the appropriate nonlinear realizations of the Euclidean symmetries of the embedding. The procedure has the advantage of giving a unified formalism with which the models are generated and highlights the relevant order parameters in each phase. We use our findings to investigate a fluid description of both tethered and hexatic membranes, showing that both the melting and the loss of local order induce long-range interactions in the high-temperature fluid phase. The results can be used to understand the appearance of intrinsic ripples in crystalline membranes in a thermal bath.