Here's a new method that I thought up today that's probably totally useless!!
Here's an example solve first (from 3x3 Example Solve Thread):
Scramble: B2 D' F R U' F2 R' D B2 R2 U2 F2 R2 U R2 B2 R2 B2 F R'
(z2 y)
B F L' D' R F2 U R2 // Step 1
F' U F2 R' F' R // Step 2
U D2 R L U L' U R' // Step 3
U F2 L2 U2 R2 B2 R2 // Step 4
(29)
The First Step is to form a "HTR F2L-1". Since you don't have to worry about the permuation of the corners yet, so you simply make a huge area of pure opposite colors. Calling it a kind of "F2L-1" limits your thinking about what you can even do with this as it's so fundamentally different from an actual F2L-1, although I will say that Petrus blockbuilding is a good framework. This first step can probably be planned in inspection and should take around 8 moves. (If you're ever using more than 11 moves, you're doing something wrong.)
The Second Step is to form an oppsite color square and orient edges, leaving only 4 corners and 3 (oriented) edges that are not in HTR. This can be achieved in a similar fashion to Tripod, but keep in mind that you do not have to preserve the opposite color area- Swapping around any edges or corners in the opposite color area from Step 1 does not mess anything up so there's some extra freedom. This is an intuitive step, doesn't take too many moves, probably one-lookable. Preferrably don't make a psuedo-opposite square/square that isn't in the back left, or else the next step's recognition could be a nightmare. The moveset is alright, but F/F2/F' moves could slow this step down a little bit.
The Third Step is the orient the remaining four corners, as well as double orient the edges and correct the corner permutation (for true HTR). This is an algorithmic step, unknown exactly how many algs but likely around 100. I still need to generate all of the algorithms, but from what I can tell, contains lots of RUD or RUL moves. Probably around 10-11 moves on average, and the example solve's case was only 7 moves long. Recognition isn't exactly great, definitely worse than something like CFOP OLL, but it's probably better than you think.
The Fourth Step is the long awaited magical Half-Turn state. In my example solve I included the one move before you actually enter this stage, because you can either do a clockwise or counterclockwise turn, depending on which one is better. The general way of reaching the finish line from here is either do a few moves to a pattern that you know and finish quickly, blockbuild your way to victory, solve the corners along with a bunch of edges and finish with a 3-cycle, or all of the above. If you have experience with FMC and DR/HTR solutions in FMC, then you will already have a pretty intuitive grasp of how to do this step efficiently and quickly. There are no algorithms in this final step, but some useful cases to be super familiar with are M' U2 M U2 and R2 U2 R2 U2 R2 U2 and U2 R2 F2 R2. If you're taking more than 13 moves (slice turn metric) or more than 4 seconds on this step, you're doing something wrong.
Here is another example solution for more comprehension of this method:
Scramble: B2 U B2 D' L2 U L2 R2 B2 R D' L' D' U' F U L2 R D
This method sucks // no (0/0)
REALIZATION: This sucks lmoa
Maybe would be better if I changed up the 2nd and 3rd steps to perhaps all corners -> all edges, would be two algorithmic steps instead of one which is probably better, and actually would be a pretty low number of algs due to the half-turn state's nature. Regardless, building the initial opposite color area is often bad, and this is just overall terrible for speedsolving even with this improvement.