Red-blue intersection requirements
I guess it depends on the particular algorithm you have in mind. However, the red/blue segment intersection algorithms that I'm aware of require that there are no monochromatic (red/red or blue/blue) intersections. For example, Chans red/blue intersection algorithm builds a trapezoidal decomposition on the blue segments. That does not really work if there are blue/blue intersections as well. 

By the way, note that both red/blue segment intersection, as well as  general (monochromatic) segment intersection can be solved in optimal $O(n \log n + k)$ time.