We study the phase structure of a phantom tethered surface model shedding light on the internal degrees of freedom (IDOF), which correspond to the three-dimensional rodlike structure of the lipid molecules. The so-called tilt order is assumed as IDOF on the surface model. The model is defined by combining the conventional spherical surface model and the XY model, which describes not only the interaction between lipids but also the interaction between the lipids and the surface. The interaction strength between IDOF and the surface varies depending on the interaction strength between the variables of IDOF. We know that the model without IDOF undergoes a first-order transition of surface fluctuations and a first order collapsing transition. We observe in this paper that the order of the surface fluctuation transition changes from first order to second order and to higher order with increasing strength of the interaction between IDOF variables. On the contrary, the order of collapsing transition remains first order and is not influenced by the presence of IDOF.