Shading Intersection Region

Possible solution

P0 = {0, 0, 0}; P1 = {1, 0, 0}; P2 = {0, 1, 0}; P3 = {0, 0, 
  1}; P4 = {1, 1, 1}; P5 = {-1, -1, -1};

R[1] = ConvexHullMesh[{P0, P1, P2, P3}, Frame -> True, 
   PlotRange -> Automatic, 
   MeshCellStyle -> {{2, All} -> Opacity[.4, Red], {1, All} -> 
      Black, {0, All} -> Blue}];

R[4] = ConvexHullMesh[{P0, P1, P3, P4}, Frame -> True, 
   PlotRange -> Automatic, 
   MeshCellStyle -> {{2, All} -> Opacity[.4, Red], {1, All} -> 
      Black, {0, All} -> Blue}];

faces1 = MeshPrimitives[R[1], "Polygons"];
faces4 = MeshPrimitives[R[4], "Polygons"];

colors1 = Opacity[0.4, #] & /@ {Green, Blue, Yellow, Red};
colors3 = Opacity[0.4, #] & /@ {Red, Yellow, Green, Purple};

g1 = Graphics3D[{Transpose@{colors1, faces1}, 
   Transpose@{colors3, faces4}}, 
  PlotRange -> {{-1.2, 1.2}, {-1.2, 1.2}, {-1.2, 1.2}}, Axes -> True, 
  ImageSize -> Large, AxesOrigin -> {0, 0, 0}, 
  LabelStyle -> {12, Bold, Black}, AxesStyle -> Thick, Boxed -> True, 
  AxesLabel -> {x, y, z}, 
  Epilog -> {Inset[Style["Inter", 20, Bold], {.2, .8}]}];
rd = RegionIntersection[R[4], R[1]];

g2 = RegionPlot3D[rd];
Show[g1, g2]

intersectionfaces =  MeshPrimitives[RegionIntersection[R[1], R[4]], "Polygons"];

Graphics3D[{Transpose@{colors1, faces1}, Transpose@{colors3, faces4}, 
  EdgeForm[{Thick, Black}], FaceForm[], intersectionfaces}, 
 PlotRange -> {{-1.2, 1.2}, {-1.2, 1.2}, {-1.2, 1.2}}, Axes -> True, 
 ImageSize -> Large, AxesOrigin -> {0, 0, 0}, 
 LabelStyle -> {12, Bold, Black}, AxesStyle -> Thick, Boxed -> True, 
 AxesLabel -> {x, y, z}, 
 Epilog -> {Inset[Style["Inter", 20, Bold], {.2, .8}]}]