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Based on the screenshot, you are only interested in quotes spanning multiple lines. Here's a patten for that:

"\\(``\\(?:[^'\n]\\|'[^'\n]\\)*\\)\n\\(?:.\\|\n\\)*?''"

Or in rx syntax:

(rx (seq (group "``" (zero-or-more (or (not (in "\n'"))
                                       (seq "'" (not (in "\n'"))))))
         "\n" (minimal-match (zero-or-more anything)) "''"))

Group 1 is the part to highlight. e.g. with hi-lock-mode:

Hi-lock: (("\\(``\\(?:[^'\n]\\|'[^'\n]\\)*\\)\n\\(?:.\\|\n\\)*?''"
           (1 (quote hi-green-b) prepend)))

Based on the screenshot, you are only interested in quotes spanning multiple lines. Here's a patten for that:

"\\(``\\(?:[^'\n]\\|'[^'\n]\\)*'?\\)\n\\(?:.\\|\n\\)*?''"

Or in rx syntax:

(rx (seq (group "``" (zero-or-more (or (not (in "\n'"))
                                       (seq "'" (not (in "\n'")))))
                (zero-or-one "'"))
         "\n" (minimal-match (zero-or-more anything)) "''"))

Note that we handle single ' characters with this:

"\\(?:[^'\n]\\|'[^'\n]\\)*'?

Which says match zero-or-more repetitions of:

1. Any single character which is neither ' nor a newline.
2. A ' character followed by any single character which is neither ' nor a newline.

Followed optionally by a single ' immediately before the newline.

Group 1 is the part to highlight. e.g. with hi-lock-mode:

Hi-lock: (("\\(``\\(?:[^'\n]\\|'[^'\n]\\)*'?\\)\n\\(?:.\\|\n\\)*?''"
           (1 (quote hi-green-b) prepend)))

I can now see the basic problem is that I want to capture from `` (two back ticks) to the end of the line, but only when there are not two consecutive quote marks (i.e., closing quote is '').

"``.*?\\(?:''\\|$\\)"

or equivalently:

(rx (seq "``"
         (minimal-match (zero-or-more not-newline))
         (or "''" eol)))

Based on the screenshot, you are only interested in quotes spanning multiple lines. Here's a patten for that:

"\\(``\\(?:[^'\n]\\|'[^'\n]\\)*\\)\n\\(?:.\\|\n\\)*?''"

Or in rx syntax:

(rx (seq (group "``" (zero-or-more (or (not (in "\n'"))
                                       (seq "'" (not (in "\n'"))))))
         "\n" (minimal-match (zero-or-more anything)) "''"))

Group 1 is the part to highlight. e.g. with hi-lock-mode:

Hi-lock: (("\\(``\\(?:[^'\n]\\|'[^'\n]\\)*\\)\n\\(?:.\\|\n\\)*?''"
           (1 (quote hi-green-b) prepend)))

I can now see the basic problem is that I want to capture from `` (two back ticks) to the end of the line, but only when there are not two consecutive quote marks (i.e., closing quote is '').

"``.*?\\(?:''\\|$\\)"

or equivalently:

(rx (seq "``" (minimal-match (zero-or-more not-newline)) (or "''" eol)))

I can now see the basic problem is that I want to capture from `` (two back ticks) to the end of the line, but only when there are not two consecutive quote marks (i.e., closing quote is '').

"``.*?\\(?:''\\|$\\)"

or equivalently:

(rx (seq "``"
         (minimal-match (zero-or-more not-newline))
         (or "''" eol)))