The incidence matrix of the real encasing polytope (ex) is

120 ♦   2   2   2   2   2   2 |   3   3   3   3   3   3   3   3   3   3 |   4   4   4   4   4
----+-------------------------+-----------------------------------------+--------------------
  2 | 120   *   *   *   *   * |   1   1   1   1   1   0   0   0   0   0 |   1   1   1   1   1
  2 |   * 120   *   *   *   * |   1   1   0   0   0   1   1   1   0   0 |   1   1   1   1   1
  2 |   *   * 120   *   *   * |   1   0   1   0   0   1   0   0   1   1 |   1   1   1   1   1
  2 |   *   *   * 120   *   * |   0   1   0   1   0   0   1   0   1   1 |   1   1   1   1   1
  2 |   *   *   *   * 120   * |   0   0   1   0   1   0   1   1   1   0 |   1   1   1   1   1
  2 |   *   *   *   *   * 120 |   0   0   0   1   1   1   0   1   0   1 |   1   1   1   1   1
----+-------------------------+-----------------------------------------+--------------------
  3 |   1   1   1   0   0   0 | 120   *   *   *   *   *   *   *   *   * |   1   1   0   0   0  (r)
  3 |   1   1   0   1   0   0 |   * 120   *   *   *   *   *   *   *   * |   0   0   1   1   0  (o)
  3 |   1   0   1   0   1   0 |   *   * 120   *   *   *   *   *   *   * |   0   0   1   0   1  (y)
  3 |   1   0   0   1   0   1 |   *   *   * 120   *   *   *   *   *   * |   1   0   0   0   1  (g)
  3 |   1   0   0   0   1   1 |   *   *   *   * 120   *   *   *   *   * |   0   1   0   1   0  (b)
  3 |   0   1   1   0   0   1 |   *   *   *   *   * 120   *   *   *   * |   0   0   0   1   1  (v)
  3 |   0   1   0   1   1   0 |   *   *   *   *   *   * 120   *   *   * |   0   1   0   0   1  (m)
  3 |   0   1   0   0   1   1 |   *   *   *   *   *   *   * 120   *   * |   1   0   1   0   0  (c)
  3 |   0   0   1   1   1   0 |   *   *   *   *   *   *   *   * 120   * |   1   0   0   1   0  (w)
  3 |   0   0   1   1   0   1 |   *   *   *   *   *   *   *   *   * 120 |   0   1   1   0   0  (k)
----+-------------------------+-----------------------------------------+--------------------
♦ 4 |   1   1   1   1   1   1 |   1   0   0   1   0   0   0   1   1   0 | 120   *   *   *   *
♦ 4 |   1   1   1   1   1   1 |   1   0   0   0   1   0   1   0   0   1 |   * 120   *   *   *
♦ 4 |   1   1   1   1   1   1 |   0   1   1   0   0   0   0   1   0   1 |   *   * 120   *   *
♦ 4 |   1   1   1   1   1   1 |   0   1   0   0   1   1   0   0   1   0 |   *   *   * 120   *
♦ 4 |   1   1   1   1   1   1 |   0   0   1   1   0   1   1   0   0   0 |   *   *   *   * 120

The vertex figure (ike) can be considered as trigonal snub antiprism, having the bases red (r), and additionally the up-edges of the equatorial zickzack colored red too. Being rolled onto an other face, the same can be done for 9 other colors (say orange (o), yellow (y), green (g), blue (b), violett (v), magenta (m), cyan (c), white (w), black (k)), thus coloring all edges and faces exactly once. - Being a vertex figure, the ike-coloring of edges represents the colors of vertex incident faces of ex, whereas that of ike faces represents that of the opposite faces of the vertex incident cells (tet). Thus all these vertex incident tet are 4-colored. There are 6 different vertex color-combinations of edges at the vertex figure (ike); these represent edge color-combinations of triangles of ex. As can be read from the matrix above, every cell (tet) of ex has one such each. Further, this scheme can be extended to all cells of ex consistently.

The incidence matrix of the complex polytope thus is

120 |   3
----+----
  3 | 120

Generators

τ = (1+sqrt(5))/2
R0 : permutes the vertices of some (red) triangle, say span( (0,0,0,τ), (τ/2,-1/2,0,τ2/2), (1/2,0,-τ/2,τ2/2) ) 
R1 : permutes the (red) triangles of some vertex, say (0,0,0,τ)

              /  0   1   τ2  τ  \          / 0  0  1  0 \
              |  τ2  τ   0  -1  |          | 1  0  0  0 |
R0 = (1/2τ) * | -τ   τ2 -1   0  | ,   R1 = | 0  1  0  0 |
              \  1   0  -τ   τ2 /          \ 0  0  0  1 /

R03  =  1                                                       /-1   τ2  0   τ  \
R13  =  1                                                       | τ   0   τ2  1  |
R0 * R1 * R0 * R1 * R0  =  R1 * R0 * R1 * R0 * R1    = (1/2τ) * | τ2  1  -τ   0  |
                                                                \ 0  -τ  -1   τ2 /