 Solution

There are no symplectic structures on the Möbius strip. A symplectic form on a surface would be a nowhere-vanishing two-form, hence would provide an orientation. The Möbius strip is nonorientable: transport around its core reverses a transverse direction and therefore reverses any local orientation. This is incompatible with such a two-form. The argument applies whether the boundary is included or only the interior is considered.