Given a pair of conics, construct (synthetically) their shared tangent lines

There are various well-known ways to construct the common tangents to a pair of circles; this is an easy one.

I also just learned that we can use Pascal's Theorem to construct a tangent to a conic at any desired point. So the question arises: how can we construct the common tangents to a pair of arbitrary conics in the plane? The construction for two circles linked above can't be used (it seems to me) because its key idea is that the circles are images of each other under affine transformations. The conics are images of each other under projective, but generally not affine, maps.

I'm looking for straightedge-and-compass solutions, not analytic ones. Since this is a purely projective problem, straightedge alone might suffice: bonus points for such a solution!

Here's an example setup, with one of the four tangents constructed by eye (orange/green). The only difficulty lies in identifying points $E$ and $E_1$. [EDIT: Note that this question is a less general version of the present one; unfortunately it's been open for 3+ months without resolution, which doesn't make me very optimistic...]