The field line connection of a tokamak sheared magnetic field ensures a finite parallel dynamical response for every degree of freedom available in the system. In the scrape-off layer (SOL) the flux surfaces are open, and the field line connection property is broken by the presence of a Debye sheath arising where the field lines strike boundary plates, hence allowing the existence of convective cell modes for which there is no dynamical parallel response. This leads to a major distinction in terms of turbulence character between closed and open flux surface regions. We study this using three-dimensional electromagnetic gyrofluid computations. The turbulence is found to change character from an ion temperature gradient to a generic interchange type, crossing the last closed flux surface (LCFS) radially outward. The width of the transition zone is about ten ion gyroradii. Various poloidal configurations of the Debye sheaths retain this interface property but affect the interaction between the turbulence and the slowly varying, self-consistent background. The strongest effect is found in a case with sheath plates at both the top and bottom of the SOL, allowing the high-field and low-field sides of the SOL to decouple. In these sides the curvature is favourable and unfavourable, respectively. The clear asymmetry observed between these sides of the plasma is consistent with previous experimental results and makes room for future experimental qualitative comparisons, for instance, on double null configurations of the tokamak ASDEX Upgrade.