In this paper we study the requirements that must be satisfied by a fluid-structure coupling scheme, in order to obtain a dynamically consistent aeroelastic code. Both spatial compatibility and time synchronization requirements must be met, to assure that the time-marching simulations exhibit the physically correct stability behavior. Inconsistent or inaccurate implementation of the fluid-structure boundary conditions can cause the aeroelastic code to converge to an incorrect aeroelastic solution. It is shown that CFD codes based on linear interpolations for the velocities cannot be fully compatible with structural FE codes that use plate or shell elements to model the wing skin. Numerical examples are presented for three different nonlinear aeroelastic models, using Euler-based aerodynamics and two different fluid-structure coupling schemes. The results indicate that for Mach numbers in the upper transonic range, past the transonic dip, the aeroelastic solution appears very sensitive to the fluid-structure coupling scheme used. In the case of the NACA 0012 model, the two different schemes studied predicted entirely different stability behaviors.