I've been meaning to make a thread on this new "method" for a while now, so here it is. I have only shared my algorithms with a select few people to see what they think, including Jeremy Fleischmann, Breandan Vallance, and Anthony Brooks, and they also believe that this system has great potential. I guess the first part of the "RLS" system is what is most commonly known as the Winter Variation.
Winter Variation (F2LL) is a set of algs which allows you to orient LL corners while solving your 4th pair if this pair is connected. The full set of Winter Variation algs can be found here: http://absolutemind.pagesperso-orange.fr/f2ll-angl.htm
Not many people have actually taken the time to learn the 27 Winter Variation cases. But I believe they are definitely worth while. Mats Valk from the Netherlands uses one of the Winter Variation cases on his 6.41 official single, ranking him 3rd in the world.
scramble : D R2 D' F2 D' L2 U R2 D2 L' B2 D' L' U F' L U' L2 D'
X-cross : x y2 D U' R' F y R U' R' u R' (9)
2nd pair : y' U' R U2 R' U R U' R' (8)
3th pair : U' L' U L U2 L' U L (8)
WV setup : U2 R' U R U2 (5)
WV: R2 D' R U R' D R2 (7)
PLL : U' R U R' U' R' F R2 U' R' U' R U R' F' U' (16)
53 moves / 6.41 = 8.27 TPS
This is the solve that inspired me to finally learn Winter Variation, and to generate RLS algorithms. RLS is simply Winter Variation with all types of last layer edge orientations. There are 216 cases in total (this number includes the 27 WV cases). Here are a few examples of some solves during which I would use RLS:
Example 1: R2 F2 R' D2 B2 L' R' B2 U2 L D2 F' U' L2 F2 L D' R' B' L2 D' (cross on U)
cross: z2 D' R' y' R' U2 R' F D' (7)
first pair: R' U' R U2 R' U R (7)
second pair: y' U' R U' R' U R' U' R (8)
third pair: y' U R' U' R U2 R' U R (8)
setup: U R U' R' (4)
RLS: y' U R' F' U' F U' R U R B' R' B (12)
PLL: U' R2 u' R U' R U R' u R2 B U' B' U' (14)
total moves: 60
Example 2: L D2 U2 L2 D2 R D2 U2 B2 L B2 U' F' D B U' B2 U2 L' D2 F (cross on U)
cross: z2 y L U2 D' R' D F (6)
first pair: L F' L' F R U' R' (7)
second pair: y R U' R' U2 R U' R' (7)
third pair: y U2 R U R' (4)
setup: y U' R U' R' (4)
RLS: U F' U F R U' R' (7)
PLL: R' U R' U' R' U' R' U R U R2 (11)
Total moves: 46
You can find my complete set of RLS algorithms on my website here: http://rowe.cubing.net/rls/
I hope people find all of this useful. Feel free to comment, question, and leave both positive and negative opinions on the system.