On the plane two ellipses can intersect in exactly four different points.
In space two ellipsoids can meet in a pair of ellipses intersecting in exactly two different points. For example take $x^{2}+(y/4)^{2}+z^{2}=1$ and $(x/4)^{2}+y^{2}+z^{2}=1$.
However, it seems impossible for two ellipsoids to intersect in a pair of ellipses intersecting in exactly four different points.
Intuitively it seems this is due to the fact that in such case the ellipses would lie on the same affine plane in space. What is the rigorous way to prove this?