I need to clean up a badly made mesh. I would like to be able to use a non-closed mesh as a cutting tool. Basically, I want to be able to place two meshes so they are overlapping and then have edges added to both meshes that follows the seam where they overlap.
Blender’s boolean tools seem to only work with properly made manifolds. I don’t need to do boolean operations, just place a cut mark where two shapes overlap. Is there a way to do this?
Do this …
Obviously they need to be joined as a single mesh.
Press G to go into move mode, and then press Enter without moving the mesh. I moved it slightly in the animation to show I was in move mode. You can leave yours right were it is.
Delete / tidy up the overlap.
You may want to duplicate the mesh and only use parts of it after doing this. This would be a work around if you do not want to auto merge.
I tried doing this, but I’m still not getting the cut marks. Both of these meshes have been joined into a single object. I selected everything, pressed G and Enter, but the surfaces are still two separate pieces.
Depending on your real objects (the example works) you might want to try the knife project tool… two times for every object (and maybe using a copy…) …first result (from above and ortho !!):
Try selecting just one piece, not both. I only have one selected in the example - until they are joined.
I think I get it - they only merge if the faces are in the same plane. I was wanting something more like an edge is created where two meshes cut through each other.
Interesting… By pure chance I must have only ever done that with flat mesh. I too are learning today 
