  Random tilings of geometrical shapes with dominos or lozenges have been a
rich source of universal statistical distributions. This paper deals with
domino tilings of checker board rectangular shapes such that the top two and
bottom two adjacent squares have the same orientation and the two most left and
two most right ones as well.
  It forces these so-called "skew-Aztec rectangles" to have cuts on either
side. For large sizes of the domain and upon an appropriate scaling of the
location of the cuts, one finds split tacnodes between liquid regions with two
distinct adjacent frozen phases descending into the tacnode. Zooming about such
split tacnodes, filaments appear between the liquid patches evolving in a
bricklike sea of dimers of another type. This work shows that the random
fluctuations in a neighborhood of the split tacnode are governed asymptotically
by the discrete tacnode kernel, providing strong evidence that this kernel is a
universal discrete-continuous limiting kernel occurring naturally whenever we
have double interlacing patterns. The analysis involves the inversion of a
singular Toeplitz matrix which leads to considerable difficulties.
