The "canonical" way is to find a pattern (here, {Black, Thick}) that matches what the boundary is made of and extract it from the graphics object.
So given
pt = RegionPlot[{{x^3 - y^2 > 2 y &&
x^2 + y^3 > 2 x}, {x^3 - y^2 < 2 y && x^2 + y^3 > 2 x}}, {x, -1,
1}, {y, -1, .1}, PlotStyle -> {Green, Yellow},
BoundaryStyle -> {Black, Thick}];
bdy=Cases[Normal@First@pt, {Black, Thick, __}, Infinity];
Graphics[bdy]
---EDIT 2---
In diagonally reading your question, I missed the requirement for the boundary between the two.
The following will work on the particular dataset. First, you can extract the points from the bdy:
points = Cases[bdy, Line[a___] -> a, Infinity]
and you will notice that there are two components each corresponding to one region. I thought that Intersection wouldn't work for the two but as @eldo points out, it turns out it does:
bdy = First /@ GatherBy[Intersection@@points, First] (* so that there are no duplicate x coords*);
gives the boundary points which can be fitted to a model of your liking or interpolate or whatever:
fit = Interpolation[bdy, InterpolationOrder -> 1];
Plot[fit[x], {x, -1, 0},
Epilog -> {Red, PointSize -> Tiny, Point[points[[1]]~Join~points[[2]]]},
PlotStyle -> {Blue, Thick}]
