Plot3D with different colors above and below 0

Question

In the Plot3D generated by the code below, I would like to indicate the region below 0 with a different color. How can this be done?

EDIT: and can the problem be solved while keeping a look/feel like the image below?

Plot3D[If[p1 == p2, -(-1 + p1) p1, ((1 - p1) p1 (-128 ((1 - p1) p1)^(3/2) (-1 +p2)^2 p2^2 + (384 (-1 + p1)^3 p1^3)/Sqrt[1/(p2 - p2^2)] + 128 Sqrt[(1 - p1) p1] (-1 + p2)^3 p2^3 (-3 + 2 Sqrt[((-1 + p2) p2)/((-1 + p1) p1)]) + 128 Sqrt[1/(p1 - p1^2)] (3 (-1 + p1) p1 (-1 + p2)^3 p2^3 - (-1 + p2)^4 p2^4 + (2 ((1 - p2) p2)^(7/2))/Sqrt[1/(p1 - p1^2)] + ((-1 + p2)^4 p2^4)/(((-1 + p2) p2)/((-1 + p1) p1))^(3/2) + 3 (-1 + p1)^4 p1^4 Sqrt[((-1 + p2) p2)/((-1 + p1) p1)] - 6 (-1 + p1)^4 p1^4 (((-1 + p2) p2)/((-1 + p1) p1))^(3/2)) + 4 (1 - p1) p1 (-128 ((1 - p2) p2)^(5/2) - 96/(1/(p2 - p2^2))^(5/2)) + 256 (-1 + p1)^2 p1^2 (((1 - p2) p2)^(3/2) + 3/(1/(p2 - p2^2))^(3/2))))/(64 (p1 - p2)^2 ((1 - p2) p2)^(3/2) (-1 + p1 + p2)^2)], {p1, 0.5, 1}, {p2, 0.5, 1}, Exclusions -> None]

Answer 1

Add the options MeshFunctions -> {#3 &} , Mesh -> {{0}}, and MeshShading -> {Red, Blue} to get

Answer 2

Use ColorFunction with ColorFunctionScaling turned off (None).

Plot3D[If[p1 == 
   p2, -(-1 + 
      p1) p1, ((1 - 
       p1) p1 (-128 ((1 - p1) p1)^(3/2) (-1 + 
           p2)^2 p2^2 + (384 (-1 + p1)^3 p1^3)/Sqrt[1/(p2 - p2^2)] + 
       128 Sqrt[(1 - p1) p1] (-1 + p2)^3 p2^3 (-3 + 
          2 Sqrt[((-1 + p2) p2)/((-1 + p1) p1)]) + 
       128 Sqrt[1/(p1 - 
            p1^2)] (3 (-1 + 
             p1) p1 (-1 + p2)^3 p2^3 - (-1 + 
              p2)^4 p2^4 + (2 ((1 - p2) p2)^(7/2))/
           Sqrt[1/(p1 - 
               p1^2)] + ((-1 + p2)^4 p2^4)/(((-1 + p2) p2)/((-1 + p1) p1))^(3/
              2) + 3 (-1 + p1)^4 p1^4 Sqrt[((-1 + p2) p2)/((-1 + p1) p1)] - 
          6 (-1 + p1)^4 p1^4 (((-1 + p2) p2)/((-1 + p1) p1))^(3/2)) + 
       4 (1 - p1) p1 (-128 ((1 - p2) p2)^(5/2) - 96/(1/(p2 - p2^2))^(5/2)) + 
       256 (-1 + p1)^2 p1^2 (((1 - p2) p2)^(3/2) + 
          3/(1/(p2 - p2^2))^(3/2))))/(64 (p1 - p2)^2 ((1 - p2) p2)^(3/2) (-1 +
         p1 + p2)^2)], {p1, 0.5, 1}, {p2, 0.5, 1},
 PlotPoints -> 150,
 ColorFunction -> (If[#3 < 0, Lighter[Blue, 0.5], Automatic] &),
 ColorFunctionScaling -> False,
 AxesLabel -> (Style[#, 12, Bold] & /@ {Subscript[p, 1], Subscript[p, 2], 
     p}),
 Exclusions -> None]