Try for starters to reduce the far range on your camera. Try with 1000. Generally speaking, you shouldn't be having overlapping faces in your 3d scene, unless they are treated in a VERY specific way (look up the term 'decal textures'/'decals'). So basically, you have to create depth offsets, and perhaps even pre sort the objects when doing this, which all requires pretty low-level tinkering.
If the far range reduction helps, then you're experiencing a lack of precision (depending on the device). Also look up 'z fighting'
UPDATE
Don't overlap planes.
How do I tell the renderer to put one of the objects in front?
You put one object in front of the other :)
For example if you have a camera at 0,0,0 looking at an object at 0,0,10, if you want another object to be behind the first object put it at 0,0,11 it should work.
UPDATE2
What is z-buffering:
http://en.wikipedia.org/wiki/Z-buffering
http://msdn.microsoft.com/en-us/library/bb976071.aspx
Take note of "floating point in range of 0.0 - 1.0".
What is z-fighting:
http://en.wikipedia.org/wiki/Z-fighting
...have similar values in the z-buffer. It is particularly prevalent with
coplanar polygons, where two faces occupy essentially the same space,
with neither in front. Affected pixels are rendered with fragments
from one polygon or the other arbitrarily, in a manner determined by
the precision of the z-buffer.
"The renderer cannot reposition anything."
I think that this is completely untrue. The renderer can reposition everything, and probably does if it's not shadertoy, or some video filter or something. Every time you move your camera the renderer repositions everything (the camera is actually the only thing that DOES NOT MOVE).
It seems that you are missing some crucial concepts here, i'd start with this:
http://www.opengl-tutorial.org/beginners-tutorials/tutorial-3-matrices/
About the depth offset mentioned:
How this would work, say you want to draw a decal on a surface. You can 'draw' another mesh on this surface - by say, projecting a quad onto it. You want to draw a bullet hole over a concrete wall and end up with two coplanar surfaces - the wall, the bullet hole. You can figure out the depth buffer precision, find the smallest value, and then move the bullet hole mesh by that value towards the camera. The object does not get scaled (you're doing this in NDC which you can visualize as a cube and moving planes back and forth in the smallest possible increment), but does translate in depth direction, ending up in front of the other.