In my viewport, some faces look like they are either disconnected from the rest of my mesh by means of duplicate vertices or edges, or they look like their normals are opposite to the rest of the mesh.

enter image description here

But the tools for fixing these problems fail to find such problems. According to these tools, there's no duplicates to merge, and the normals ARE consistent.

What other strange geometry problems could make my faces look inconsistent in my viewport?

Added a share of my file as requested:

  • 4
    $\begingroup$ Please use the edit link at the bottom of your question (i.stack.imgur.com/lXFuK.png) and add more information on how you set up your project. Add images that might help us understand your scene and settings. See How to upload an image to a post? $\endgroup$ – user1853 Mar 8 '19 at 7:49
  • $\begingroup$ maybe share your file: blend-exchange.giantcowfilms.com $\endgroup$ – moonboots Mar 8 '19 at 10:35
  • $\begingroup$ Visual problem with the normals is probably due to the length of the normals, so that they intersect with faces. Just decrease the size of the normals to get a better vision $\endgroup$ – Ocaso Protal Mar 8 '19 at 11:42

Some of your faces are shaded "smooth" and others "flat", select your mesh and hit either flat or smooth under the Tool-Menu -> Shading

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