This paper presents a new type of surface models constructed on the basis of Finsler geometry. A Finsler metric is defined on the surface by using an underlying vector field, which is an in-plane tilt order. According to the orientation of the vector field, the Finsler length becomes dependent on both position and direction on the surface, and for this reason the parameters such as the surface tension and bending rigidity become anisotropic. To confirm that the model is well-defined, we perform Monte Carlo simulations under several isotropic conditions such as those given by random vector fields. The results are comparable to those of previous simulations of the conventional model. It is also found that a tubular phase appears when the vector field is constant. Moreover, we find that the tilts form the Kosterlitz-Thouless and low temperature configurations, which correspond to two different anisotropic phases such as disk and tubular, in the model in which the tilt variable is assumed to be a dynamical variable. This confirms that the model in this paper may be used as an anisotropic model for membranes.