You're describing Z-fighting. Add an Offset to the shader used for one of the particle emitters. Just copy the entire shader, and add the single line of code needed for offset. Offset -1, -1 will make the particles render in front, and Offset 1, 1 will make them render behind.
Shader "Particles/Alpha Blended" {
Properties {
_TintColor ("Tint Color", Color) = (0.5,0.5,0.5,0.5)
_MainTex ("Particle Texture", 2D) = "white" {}
_InvFade ("Soft Particles Factor", Range(0.01,3.0)) = 1.0
}
Category {
Tags { "Queue"="Transparent" "IgnoreProjector"="True" "RenderType"="Transparent" }
Offset -1, -1 // This line could be here, or anywhere else before "SubShader".
Blend SrcAlpha OneMinusSrcAlpha
// ...