Hat tilings with given Conway signatures

In what follows I will call cluster H7 (resp. H8) tile 7 (resp. tile 8) and rely on the fact that any tiling of the plane using tiles "7" and "8" is the (unique) result of a subdivision of another, coarser, tiling by tiles "7" and "8" (the transition from the fine mesh to the coarse mesh is called inflation.
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Given a tile ("7" or "8") of a tiling, its Conway signature is an infinite-to-the-left sequence of symbols taken in {0,1,2,3,4,5,6} that records the subdivision history that originated it. Most recent subdivision history is recorded to the right, in particular the last symbol of the signature is 0 iff the are focusing on a tile "7".
The meaning of the symbols is clarified in the image on the left showing the subdivision process for a tile "8".
In a signature we do not allow for the consecutive pair of symbols ...06...
The notation "[w]p." denotes the periodic signature "...wwwwp." where w is any nonempty finite sequence of symbols and p is a possibly empty finite sequence of symbols.
A tile is part of an infinite worm if and only if its signature contains only symbols in {0,2,6};
An infinite worm is described as a bi-infinite sequence of tiles "8" and "7" conventionally oriented as illustrated.
Hereafter we show the neighborhood of radius 6 of a tile in a tiling by its Conway signature
Overview Zoom Double zoom Signature
[0].
partial tiling - restriction of a bi-infinite worm
The sequence of "7" and "8" corresponds under 7->0, 8->1 to the infinite-to-the-right string
0r
where r=101101011011010110101... is the rabbit string with rabbit signature [1].
This is the part of the yellow brick road that Dorothy walks in the povray animations.
[6].
partial tiling - restriction of a bi-infinite worm
The sequence of "7" and "8" corresponds under 7->0, 8->1 to the infinite-to-the-left string
m(r)1
where m(r) is the mirror image of r. This is the part of the yellow brick road behind the starting position of Dorothy.
[2].
partial tiling
infinite worm The sequence of "7" and "8" corresponds under 7->0, 8->1 to the bi-infinite string
m(r)01r
having rabbit signature [01].
[36].
partial tiling of period-2
[63].
partial tiling of period-2
[026].
infinite symmetric worm
The sequence of "7" and "8" corresponds under 7->0, 8->1 to the bi-infinite symmetric string having rabbit signature [011]. and centered at a one-one pair: ...11...
[260].
infinite symmetric worm corresponding to the symmetric rabbit signature [101]. and centered at a zero: ...0...
[602].
infinite symmetric worm corresponding to the symmetric rabbit signature [110]. and centered at a one: ...1...
[1].
an example of a tiling with no infinite worm
[3].
another example of a tiling with no infinite worm
[4].
another example of a tiling with no infinite worm
[5].
the last example of a tiling with period-1 signature
This seems to be a partial tiling
Examples with union of multiple partial tilings
Overview Zoom Double zoom Signatures
[0]. and [6]. twice
disjoint union of three partial tilings
[36]. three times
union of three disjoing copies of [36].
[36]. three times
union of three disjoing copies of [63].
[5]. twice
union of two disjoing copies of [5].