Do WHERE IN queries reindex the table after each set within the query?
No SELECT causes reindexing.  Only ALTER, OPTIMIZE, and a few other commands, may cause reindexing.
The construct (a,b) IN ((1,2), (3,4)) is not well optimized.


So,...  Make it two DELETEs, such as

DELETE FROM `table1`
    WHERE djID = 128
      AND localID = 26;


If you have lots of them, you could build a tmp table with the pairs of numbers, then do a multi-table DELETE.

To demonstrate that it is doing far more work than it ought to,

FLUSH STATUS;
DELETE ... WHERE ... IN ...;
SHOW SESSION STATUS LIKE 'Handler%';


You will probably see some numbers about 20M (or maybe 40M), indicating that it did a table scan in spite of having the index.

Then file a bug with http://bugs.mysql.com (unless there already is one).