I am trying to get Mathics3 to produce Asymptote output that more closely matches what Mathematica produces.

Here is an example of Mathematica output for Plot3D[Sin[x y], {x, 0, 3}, {y, 0, 3}]:

and here is what Mathics3 produces for the same thing:

A small portion of the file looks like this

import three;
import solids;
import tube;
size(6.6667cm, 6.6667cm);
currentprojection=perspective(1.3,-2.4,2.0);
currentlight=light(rgb(0.5,0.5,1), specular=red, (2,0,2), (2,2,2), (0,2,2));
// Polygon3DBox
path3 g=(0,0,0)--(0,0.033333,0)--(0.033333,0,0)--cycle;draw(surface(g), rgb(1, 1, 1), nullpen);
// Polygon3DBox
path3 g=(0,0.033333,0)--(0.033333,0,0)--(0.033333,0.033333,0.002)--cycle;draw(surface(g), rgb(1, 1, 1), nullpen);
...
draw(((0.0,0.0,0.14000061891403817)--(0.0,0.025,0.14000061891403817)), rgb(0, 0, 0)+linewidth(1.2));
draw(((0.0,0.0,0.16000070733032934)--(0.0,0.025,0.16000070733032934)), rgb(0, 0, 0)+linewidth(1.2));
draw(((0.0,0.0,0.18000079574662048)--(0.0,0.025,0.18000079574662048)), rgb(0, 0, 0)+linewidth(1.2));

To narrow things let us start simple, let's say this is solved if we get the color and shadowing similar.

.html#1271 is where you can find default options that Mathematica uses.

Edit: There was a bounty on this which has ended. If in the future, someone offers something interesting, I will be happy to add another bounty and award that.