FINITE PORTION OF ANSWER
The black lines show boundaries between different "crystalline forms" which can be extended infinitely.
This can be extended infinitely in all directions - see my route to solving below for how.
First a detour to explain how I made a tool (which competing answers could also use, perhaps improving on the finite number of differently-coloured tiles) by which ideas can be quickly tried using an Excel spreadsheet:
- Select all, and set column width to about 20 pixels so you get a square grid.
- Enter 'R' in K9, 'D' in M10, 'L' in O9 (this will become the starting layout as in question)
- Select a region from B2 to (e.g.) DZ100 (large enough to play with - in screenshot below, it seems I'd selected to CX63)
- Open "Conditional formatting" => "Manage rules..." dialog.
- Create 4 new rules. Each will have as its action setting one of the 4 borders of a cell. For each "Use a formula to determine which cells to format" In particular:
=OR(B2="R",AND(B1<>"",B1<>"U"),AND(B3<>"",B3<>"D"),AND(C2<>"",C2<>"R")) => format set left border
=OR(B2="L",AND(B1<>"",B1<>"U"),AND(A2<>"",A2<>"L"),AND(B3<>"",B3<>"D")) => format set right border
=OR(B2="D",AND(B3<>"",B3<>"D"),AND(A2<>"",A2<>"L"),AND(C2<>"",C2<>"R")) => format set top border
=OR(B2="U",AND(B1<>"",B1<>"U"),AND(A2<>"",A2<>"L"),AND(C2<>"",C2<>"R")) => format set bottom border
- After this, the starting cells should have the right shapes around them. I then used (non-conditional) background formatting to highlight those starting pieces.
- Play around by adding 'L', 'R', 'D' and 'U' to other cells. You can copy-paste regions around easily too...
In particular, portions of the layout that extend to infinity will likely need to follow a simple infinitely-repeating pattern such as the one shown in the answer given by PuzzlesAndSolutionsYT
After setting everything up as described above, it should look a bit like the following:
route to solving
One observation is that, within the infinitely-repeating section,
pairs of tetranimos must be "back-to-back" along their long edge, as if this were not the case, the long edge would form a 3-way junction with 2 other tiles.
At least one other possible infinite tiling pattern exists.
If, outside the finite area,
there were regions of different "crystal structure", there would need to be a join that can be extended infinitely, still using only 2 colours. This is indeed possible, using a join along a diagonal. e.g. see below a join between a L/R crystalline area and a U/D crystalline area.
Using this knowledge I was able to put together the solution at the top of the post.