Two ideas:
1) Change the order of the green and red ellipses, so the green comes on top of the red. This will produces a more greenish result where the two overlap.
2) Change to a slighlty different red or green. You are not limited to the default red, green, etc.
\definecolor{myRed}{rgb}{0.8 0.1 0.2} %for example
Luís Sequeira