You may recognise this as the Mirror Cube – but add some recognisable Rubik’s colours and this makes for an exciting new addition to the range. Simply rotate the medge that needs flipping into the unsolved layer and perform the double edge flip as described above. Then the vertical slice is rotated once more but the bottom slice on the fourth rotation is turned opposite to the previous three times.

Rubiks Blocks - Etsy UK Rubiks Blocks - Etsy UK

When only 2 edges need flipping I orient them so they are in the bottom opposite one another, one in the bottom front of the cube, the other at the back.If you watch the piece this takes the two of them, splits them rotates them, and aligns them reorientated on the top layer. With 4 flipped edges I rotate the middle vertical slice 90 degrees, rotating the bottom edges into the front. I rotate the medge piece into the final layer and in the process rotate a placed/oriented piece out of the final layer. I would suggest starting to look ahead (which you might do earlier than this, but could do here), you know you will want all the top (as you show) edges to have the correct flip, this would be an ideal point at which to start combining your efforts - your method most likely does the corners next, but it is most likely that the manoeuvres you already use do not flip the edges*. The 4 vertical center turns are repeated and the bottom turns repeated, 3 one direction and the 4th one back.

rubiks cube - Flip edge in Equator layer - Puzzling Stack rubiks cube - Flip edge in Equator layer - Puzzling Stack

The Block's unique mechanism and different sized blocks result in the Cube changing shape with every move you make. I personally put a random buffer-piece in and then solve it, the thing you want to prevent (hence this comment instead of an answer). My advice is that if you don't have any empty slots, then you might as well continue doing what you're doing. You now need to rotate the moved middle edge back into the middle slice and in the process the relocated bottom edge back into the lower slice.This is in fact only one formula, but the others are the inverse and mirror image along the FR-plane. All of the Mirror Cube's stickers have the same color (traditionally reflective silver or gold - thus its "mirror" like look) but can be difrenciated by size.

Rubik’s Blocks, Original 3x3 Cube with a Twist, Challenging

Either way, once you learn a more advanced method, the solution to this problem would be to avoid being backed into a corner with this case at all. This takes the two pieces, rotates them to the back, pulls the right side back in place, flips the front for the two pieces, and then rotates them in and flips it down. This is done to create either 4 flipped edges, in the bottom layer, or 2 flipped edges, opposite one another. If the first 4 bottom slice turns were 3 clockwise and one counterclockwise, the next 4 need to be 3 counterclockwise turns followed by a clockwise one. While having the exact same mechanism, the Rubik's Cube parts all have the same size but differ from one another with the help of colorful stickers, the Mirror Cube is the exact opposite.

But if you want something that doesn't touch the F2L at all AND is quick to execute, then you'll have a harder time finding them. Solving the Mirror Cube is exactly the same as solving the Rubik's Cube, however it is harder to identify each part because the size difference between the puzzle's parts is less obvious than the color difference of the original puzzle's stickers. Whatever moves were employed to rotate the flipped medge and position it opposite the flipped bottom edge piece need to be reversed.

Cubesolver

A way to solve this is to replace the flipped edge by a random block and then put the edge at its right place. The algorithm you've provided doesn't address the case listed in the answer above, where every edge is solved except one. Additionally, it is not nearly as efficient as simply removing the corner/edge pair, since you'd have to do this anyway to solve F2L from this position.An intuitive solution is to remove both the corner and edge, splitting them up, then move them back in an intelligent way. Once the bottom edge pieces are put in place, either one or three edge pieces are flipped, due to the flipped medge. While this does flip the single edge over, this also moves the corner out of place, and as a result, is not actually a solution to the OP's problem.