Extracting nonorthogonal sources in ICA/PCA/blind source seperation problem

My problem is essentially a 'blind source separation' problem. I have 3 non-orthogonal sources (or basis functions) and N random linear combinations (mixes) of said sources. My problem is to obtain the sources from the mixes.

Figure A shows the sources, B shows the mixes.

Approaches taken: 1) PCA- I tried PCA on the mixes (fig.C), but the issue is that PCA will only give orthogonal bases, while my sources/bases are non-orthogonal. This issue with PCA is shown in figure D, where the data is clearly described by 2 non-orthogonal basis, but PCA (the solid lines) cant reconstruct them!

2) Factor Rotation - I tried applying some solutions form Factor Analysis (not my forte). The promax rotation (matlab: nw = rotatefactors(cov,'method','promax') ) is shown in fig. D with the dashed lines. As far as I can tell, factor rotation works with the principal components, not the original matrix and thus I have no idea how it could reconstruct the right basis. I think this only works with factor matrices, not with generic ones...

3) ICA - I tried overdetermined ICA with the fastICA algorithm (also not my forte, but I think I understand it better). I was hoping this would work since independent components (ICs) are non orthogonal. The solution is shown in fig.E. While the ICs are indeed nonorthogonal, they are NOT my original sources :-(

Any other potential tips or leads or solutions would be greatly appreciated.

Thanks!