plotting – RegionPlot3D over non-rectangular range

I want to make a RegionPlot3D only in the range x + y + z < 1. The following works, but is slow and poor quality.

ρ12 = ρ23 = ρ31 = 0.4;
a12 = a23 = a31 = 2;

RegionPlot3D({
  1/ρ12 > a12 x/y > ρ12 && x + y + z < 1,
  1/ρ23 > a23 y/z > ρ23 && x + y + z < 1,
  1/ρ31 > a31 z/x > ρ31 && x + y + z < 1
  }, {x, 10^-10, 1}, {y, 10^-10, 1}, {z, 10^-10, 1},
 PlotPoints -> 101, PlotStyle -> Opacity(0.5), Mesh -> None, ViewPoint -> {1, 1, 1})

enter image description here

Any ideas on how to make this sharper looking (highest priority) and faster (lower priority)? I’m not wedded to using RegionPlot3D if there are approaches using different functions.

Bonus question: can I make the faces towards the viewer a different style?