Theorem(G. Megyesi, F. Sottile and Th. Theobald)
Fix 2 skew lines L1 and L2 and a general quadric Q in 3-dimensional projective space.
The closure of the set of quadrics for which there are infinitely many lines simultaneously transversal to L1 and L2 and tangent to both quadrics is a curve in the 9-dimensional projective space of quadrics. This curve consists of 12 plane conics.
Remark
Idea of Proof:
The case of two lines meeting in a point reduces the question to a
planar problem.