This problem occurs when loading a prefab dynamically, that was previously exported as an AssetBundle.
I export the scene and the AssetBundle as a WebPlayer build, all according to the usual practices. In fact the loading of the prefab and rendering the cube with texture A works fine!
Here comes the problem: The reference of texture A in the MonoBehaviour script is still missing, even though the same asset is used to render the cube inside of the prefab. This means the asset is loaded in memory, but for some reason the inspector reference is broken.
I can provide a simple test scene to replicate this. Sounds like a bug to me, but would appreciate feedback :)