We develop a method for solving elliptic partial differential equations on surfaces described by CAD patches that may have gaps/overlaps. The method is based on hybridization using a three-dimensional mesh that covers the gap/overlap between patches. Thus, the hybrid variable is defined on a three-dimensional mesh, and we need to add appropriate normal stabilization to obtain an accurate solution, which we show can be done by adding a suitable term to the weak form. In practical applications, the hybrid mesh may be conveniently constructed using an octree to efficiently compute the necessary geometric information. We prove error estimates and present several numerical examples illustrating the application of the method to different problems, including a realistic CAD model. •A robust method for elliptic PDE on multipatch CAD surfaces with gaps/overlaps.•Patches are weakly coupled via a hybrid variable on a 3D background grid.•The hybrid variable field is stabilized orthogonally to the interface curves.•Trimmed patches are facilitated using CutFEM stabilization techniques.