grid deduction - Tetromino Sudoku

^{An entry in Fortnightly Topic Challenge #32: Grid Deduction Hybrids}

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The grid below, when filled in, forms a valid Sudoku grid. It can also be filled in like a LITS (nuruomino) puzzle without the 1x4 tetrominos.

The two objectives of this puzzle are:

• Fill in the numbers in the grid.


• Identify the LITS tetrominos.

In addition to the standard rules of Sudoku and LITS, the following rules apply:

• Each 3x3 region in the Sudoku grid contains exactly one tetromino, which is either a L, T, or S tetromino.


• The tetrominos obey the rules of a LITS puzzle.


• The shaded yellow spaces are part of a tetromino.


• Each tetromino contains four numbers that add up to 20.


• No two tetrominos may contain the same set of four numbers.


• The tetrominos (each in its own 3x3 region) are arranged in a 3x3 Sudoku, using tetromino shapes L, T, and S instead of numbers. If two regions contain the same tetromino shape, then they cannot share the same horizontal alignment or vertical alignment.

Good luck!

Answer

I believe I have found multiple solutions to the puzzle. It is possible that there is a nuance to the rules that I missed that disqualifies one or both of them. My understanding of the rules did not allow for a full logical deduction of the solution, so these were both found by brute force.

The sudoku is on the left, while the tetrominoes are on the right:

951 837 426 42
376 542 918 37 4 9
428 916 573 28 916 5

532 678 149 3 6 4
147 329 685 47 329 68
869 154 237 6 2

683 491 752 83 91 752
794 285 361 4 28 6
215 763 894 5

The lower-center tetromino can be flipped to:

. 1
. 85
. 6

hence the multiple solutions.