References and upper bounds for the SONNAT tiling game?

Question

Introduction

In a video released about a month ago, Pembesita describes¹ a tiling game called SONNAT: Same Orientation Neighbour Not Allowed, Tiling.

In the single-player game², the player may employ two rhombi. The first rhombus has inner angles with 72, 108, 72, and 108 degrees, respectively. It has a different colour for each of its 360/36 = 10 orientations. The second rhombus has inner angles with degrees 36, 144, 36, and 144 degrees. It always has the same colour, regardless of its orientation: black, in this case.

                                           

Rules

There are two rules for the game. The first one is that no two rhombi with the same colour (orientation) are allowed to be joined to one another along their sides. The second is that no "repeating patterns" may be formed with the tiles. Below are two examples of such repeating patterns. These repeating patterns are of length two. Repeating patterns of a higher length aren't allowed either.

                                       

The goal of the game is to tile the plane with as many rhombi as possible, while respecting these rules. It is not quite clear how big the tilings can get, let alone whether it's possible to tile the whole plane.

Here is, for instance, a tiling with 67 rhombi, 57 of which are contained within a circle. It is not entirely clear how one should proceed to make a tiling with as many tiles as possible.

                          

Penrose tiles

As noted by RavenclawPerfect, the Penrose P3 aperiodic tiling is also formed by means of these two tiles. However, the tilings created in the SONNAT tiling game adhere to more strict rules. So even though they might look somewhat similar, they are not the same in terms of self-similarity and the kinds of patterns that are allowed to emerge. Compare and contrast the Penrose P3 tiling below with the bounded SONNAT tiling above:

                                       

One can see that in the P3 tiling, there are many instances of the second rhombus being adjacent to itself, thereby violating the first rule of the game.

Questions:

1. Has this game appeared in the literature before?
2. Are there any upper bounds known (or can they be derived) for the number of tiles one can reach in this game?
3. If an infinite amount of tiles can be employed, is it possible to tile the whole plane? Or only a subset of it? If the former is the case, what strategy should one employ to tile $\mathbb{R}^{2}$ ? If the latter is the case, what kinds of infinite subsets can be constructed?

Notes

1 The code for different variations of the tiling game can be found on this page. The particular variation of the above game is described in Sonnat_Or10.py. All credits to Pembesita.

2 The word "game" in this context ought to be understood in a colloquial sense. Perhaps a better term is a puzzle, similar to the one described over here. It doesn't have anything to do with game theory.