Well, a pleasant surprise! It _is_ possible to take a set of 24 distinct
tiles from any Rubik's Tangle, and use them to tile the surface of a 2x2x2
cube such that all touching ropes match. And the solution is unique!

I'll include the solution below, after some blank lines to avoid spoiling it
for anyone who wants to try solving the puzzle without seeing the answer...

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First, a hint. If you look at any face of the cube, and look at the two
pairs of colors at any edge of that face, the two pairs will be the same.
That is, if one tile touches a cube edge with colors red-blue, the adjacent
tile on that face touching the same edge will also touch the edge with red-
blue. I see no obvious reason why the solution should have this property,
but it does.

Solution below.

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Solution: Note that there is also a correlation between the color pairs that
occur within a face, and the color pairs that occur at the edges of that face.
Also, the orientation of every tile is the same relative to the adjacent of
the cube that the tile touches. This makes it relatively easy to reconstruct
the solution manually.

 BR   BR                     
G..G G..Y                    
R..Y Y..B                    
 YB   GR                     

 YB   GR                     
G..R R..Y                    
R..B B..B                    
 YG   YG                     

 GR   GR   YG   YG   BY   BY   RB   RB 
Y..Y Y..B B..B B..R R..R R..G G..G G..Y
R..B B..G G..R R..Y Y..G G..B B..Y Y..R
 BG   YR   RY   BG   GB   RY   YR   GB 
                                       
 BG   YR   RY   BG   GB   RY   YR   GB 
Y..R R..B B..G G..R R..Y Y..G G..B B..Y
R..G G..G G..Y Y..Y Y..B B..B B..R R..R
 BY   BY   RB   RB   GR   GR   YG   YG 

 RB   RB                     
Y..Y Y..G                    
B..G G..R                    
 GR   YB                     

 GR   YB                     
Y..B B..G                    
B..R R..R                    
 GY   GY