Looks like a compound of 2 coincident rectified tesseracts (rit) with 64 further tetrahedra (tet) which increase the so far double coincidences to quadruple coincidences.

Incidence matrix according to Dynkin symbol

o3o3β4β

both( . . . . ) | 64 |   3   3 |  3  3  3   9 |  1 3  4
----------------+----+---------+--------------+--------
sefa( . o3β . ) |  2 | 192   * |  1  0  1   1 |  1 1  1
sefa( . . s4s ) |  2 |   * 192 |  0  1  0   2 |  0 2  1
----------------+----+---------+--------------+--------
      . o3β .   ♦  3 |   3   0 | 64  *  *   * |  1 1  0
both( . . s4s ) ♦  4 |   0   4 |  * 48  *   * |  0 2  0
sefa( o3o3β . ) |  3 |   3   0 |  *  * 64   * |  1 0  1
sefa( . o3β4β ) |  3 |   1   2 |  *  *  * 192 |  0 1  1
----------------+----+---------+--------------+--------
      o3o3β .   ♦  4 |  12   0 |  4  0  4   0 | 16 *  *
      . o3β4β   ♦ 24 |  24  48 |  8 12  0  24 |  * 8  *
sefa( o3o3β4β ) ♦  4 |   3   3 |  0  0  1   3 |  * * 64

starting figure: o3o3x4x