Расскажу, что знаю I
Tropical Geometry is all about tori. An algebraic torus is seen as the natural complexifiation of the total space of a tangent bundle to a topological torus. This comes with the trivality of the budndle, say by the translation by left=rigt group action of the base. The coamoeba map is just a the projection to the base which we also call the phase. Tropicalisation is about degenerating along the fibers using the homothety action.

No we suggest to replace the torus with another Lie group. The first one comming to mind is SU(2), homeomorphic to the (3 dimensional) sphere in quaternions,or its double cover SO(3) -- , topologycally being RP3.

After the complexification these groups manifest as SL(2,C) and PSL(2,C). The first group geometrically is a smooth affine complex quadric in C4. The second one is RP3 with Q={determinant=0}=CP1xCP1 in CP3 removed.

The amoebas and coamieba map come from the polar decomposition on the corresponding group.
I describe the amoebas of lines. Those which are tangent to Q become hrosheres. Generic to Q lines become geodesic cylindres.

Excersise 0. Take a surface S in CP3 different from Q, then the complements to the compactified amoeba is convex.

Hin: We mean that the amoeba map is given by A->AA* and takes values in the Hyperbolic three sapcethe, the liqht cone in Hermitian 2x2 matrices with (3,1)-metric computed as the determinant.