Solution:

Notation:

Hold the puzzle with one face on top, its three corners pointing back, left, and right. Let U denote this top face. Let L, F, and R denote the left, front, and right visible faces. Note that F face shares a whole edge with U, whereas L and R each only share a vertex piece with U (and also a vertex with each other). The D face is the face on the bottom opposite the U face.
Clockwise turns of a face layer are denoted by the appropriate letter, anti-clockwise turns by the letter with an apostrophe (i.e. U', L', F', R', or D').

Phase 1: Solve the edges
This is quite straightforward. It is equivalent to solving a Dino Cube. The main difference is that you have to keep checking the colour arrangement on the vertex pieces to verify what the colour arrangement of the edges should be.

1. Choose a colour. Put the three edges with that colour together so that they have that colour together on one face. This is quite easy except that they must be placed in the correct order. Check that they are in the right order by looking at the colour arrangements on the three vertex pieces that they share the colour with.
2. With the three solved edges in the U face, solve the middle layer edges. You can do this without moving the three previously solved edges at all. Inserting an edge from the bottom layer into the middle layer is done as follows:
   1. DL -> FR: Do the move sequence: R' D R
   2. DR -> FL: Do the move sequence: L D' L'
   You are free to turn the D layer, which allows you to insert any edge in the D layer into the middle layer. You can also use either sequence to remove any edge from the middle layer by displacing it by another. In this way all the middle layer edges can be solved.
   Remember to keep checking the colour scheme is correct using the vertex pieces.
3. The last three edges are now trivially solved by at most a single turn of the D layer.

Phase 2: Solve the vertex pieces
This phase is equivalent to solving the edges of a Pyraminx.

1. Put all the vertex pieces in their correct locations (but not necessarily the correct orientation) by using the following three-cycles:
   1. UR -> UL -> LR: Do the move sequence: R U' R' U
   2. UB -> UR -> LR: Do the move sequence: R U R' U'
   You can always hold the puzzle such that one of these sequences will correctly position at least one vertex and not disturb any already placed ones. Repeat this until they are all placed correctly.
2. You can flip an adjacent pair of vertex pieces using the following sequence:
   UR+, UL+: Do the move sequence: R' U L' U' L U' R U, or alternatively do R U' R' U R' L R L'
   If two opposite (non-adjacent) vertices need to be flipped, then you could do any move to make the two vertex pieces adjacent, flip them with the move sequence, and then undo the first move.

Phase 3: Solve the centre pieces

1. Choose any unsolved centre piece. Hold the puzzle such this piece lies in the L face and that the face it belongs is the U face.
2. Turn the L face so that its incorrect centre piece (of the U face colour) lies on the right (in the r layer).
3. Turn the U face so that one of its incorrect centre pieces lies on the right (in the r layer). If you have a choice, prefer a centre piece of the L face colour over any others.
4. Do r U r' U' five times. This affects only centre pieces.
5. If necessary, turn the U face back the way it was, undoing the move from step c.
6. If necessary, turn the L face back the way it was, undoing the move from step b.
7. Repeat steps a-f until all the centres are correct.