Step by step deduction
Firstly, note that MASTERING is a full nine-letter word so it takes up a whole row, and EMIGRANT is an eight-letter word so the column is either EMIGRANTS or SEMIGRANT. Also note that ARTEMIS must begin from either the 1st or 3rd place in its row, because otherwise the A will clash with MASTERING; and the remaining letters in that row will be N and G.
If the long words MASTERING and EMIGRANT do not intersect, then there are only two possibilities:
(ARTEMIS must start from the 1st or 3rd of the row, so there's only one place it can be. Then STRANGE must have its N or G in the ARTEMIS row, so it must be in the 1st column, and we get a contradiction from two R's in the same box.)
(Same as before, only one place for ARTEMIS, then STRANGE must be in the 1st column and we get a contradiction from two A's in the same box.)
So the long words MASTERING and EMIGRANT must intersect, and there are sixteen ways that can happen.
One/two, three, four, five/six, seven/eight, nine, ten, eleven lead to just one possibility for ARTEMIS and then a quick contradiction.
- One leads to two possibilities for ARTEMIS, both leaving no possibility for STRANGE.
- Another one, again two options for ARTEMIS, but the second one is actually viable! We'll come back to that.
- Another one, ditto two options, but again the second one is actually viable.
- Another one leads to a full three different possibilities, but all of them give no possible option for STRANGE.
- Another one gives almost the same three options, again with no possibility for STRANGE.
Now we've reduced the whole playing field to just two possible Sudoku puzzles to be solved:
This one has no possible place for ENIGMA.
So we proceed with the first of these. There's only one possible place for ENIGMA, and after that only one possible place for ANIME:
After some more pure Sudoku-ish deductions, we get to here:
Now SNAG can only be in the third row, 4th to 7th places.
And some more purely Sudoku-ish deductions lead to the final answer: