A ring plane mesh showing through the mesh it loops around?

Ok, so this is a bit of an odd situation, and maybe this is just my fault for stretching the potentiality of the Unity engine thin. I am trying to make a very stylized minimalistic space game, and I'm randomly generating some planets that are basically flat circle meshes. I'm using meshes and not sprites because I want to be able to zoom in as much as possible without it looking bad. So, I have now procedurally generated rings to surround some of the planets. These rings are themselves flat meshes, except they're circles with a hole in the middle. The planet circle lies in the XY plane, while I put the ring in the XZ plane, with the same centred but made a bit bigger than the planet.

Now what happens is, I see the ring in the proper perspective, but it all shows through! I was expecting for part of it to be hidden behind the planet. I double checked the geometry, it's right, and if I purposefully create only a half-ring (the half behind) it works correctly, so I can only assume that what's happening is that the rendering order checks the ring, somehow decides it lies "in front" of the circle overall, and doesn't realize that's true only for some of it, not for its entirety. Btw I'm using a camera in Perspective mode but with an Orthographic transparency sort mode. The ring is a mesh renderer + mesh filter object, materials are sprite-diffuse everywhere, the ring itself is semi-transparent. Right now my best solution is to split the ring into two halves, front and back, and render them separately - if they turn out to look seamless that should work fine. Otherwise I have no clue. Thanks for your help!