Swordfish
Three rows locking a value to three columns clear those columns
How to use it
The three-line fish. For one symbol, find three rows whose homes all lie within the same three columns (each row needs one to three of them, as long as the union is exactly three). The three rows must then deliver the symbol into those three columns — once each — so every other occurrence of the symbol in the three columns is impossible.
Swordfish patterns need sparse candidates: a symbol with two or three homes per row across three rows, and columns that overlap. The same table used for the x-wing, read one row further, reveals them; run it on both layers.
Why it is sound
Suppose a symbol's homes in rows r1, r2, r3 all lie within columns {c1, c2, c3}. Each row contributes one occurrence and no column may receive two, so the three rows deliver the symbol exactly once into each of the three columns. Every other candidate of the symbol in those columns is impossible. Sound for the same pigeonhole reason as the x-wing. The x-wing's row/column roles transpose; only row-based swordfish is covered here — column-based is the same idea with rows and columns swapped.
A worked example
The three-line fish on the letter A. In the position shown, row 3 leaves A only in columns 6 and 7; row 4, only columns 3 and 6; row 7, only columns 3 and 7. Three rows, three shared columns {3, 6, 7}: the rows must deliver A exactly once into each of the three columns, so A is impossible elsewhere in columns 3, 6 and 7 — it falls at r1c6 and r5c3 here. Each row needs only two of the columns; what matters is that the union is exactly three. Same table as the x-wing, one row longer, on both layers.