Structural geometrical nonlinearities strongly affect the response of PrandtlPlane Joined Wings: it has been shown that linear buckling evaluations are unreliable and only a fully nonlinear stability analysis can safely identify the unstable state. This work focuses on the understanding of the main physical mechanisms driving the wing system's response and the snap-buckling instability. Several counterintuitive effects typical of unconventional non-planar wing systems are discussed and explained. In particular, an appropriate design of the joint-to-wing connection may reduce the amount of bending moment transferred, and this is shown to dramatically improve the stability properties. It is also demonstrated that the lower-to-upper-wing stiffness ratio and the torsional-bending coupling, due to both the geometrical layout and anisotropy of the composite laminates, present a major impact on the nonlinear response. How the material anisotropy modifies the Snap Buckling Region and the response is also discussed. The findings of this work could provide useful indications to develop effective aeroelastic reduced order models tailored for airplanes experiencing important geometric nonlinearities such as PrandtlPlane aircraft, Truss-braced and Strut-Braced wings and sensorcrafts.