(t
        (setq p-address address)))
 
-      ;; Remove elements of the mailbox part that are not relevant for
-      ;; display, even if they are required during transport.
       (when p-name
-       ;; Outer double quotes.
-       (when (string-match "^\"\\(.*\\)\"$" p-name)
-         (setq p-name (match-string 1 p-name)))
-
+       ;; Remove elements of the mailbox part that are not relevant for
+       ;; display, even if they are required during transport:
+       ;;
        ;; Backslashes.
-       (setq p-name (replace-regexp-in-string "\\\\" "" p-name)))
+       (setq p-name (replace-regexp-in-string "\\\\" "" p-name))
+
+       ;; Outer single and double quotes, which might be nested.
+       (loop
+        with start-of-loop
+        do (setq start-of-loop p-name)
+
+        when (string-match "^\"\\(.*\\)\"$" p-name)
+        do (setq p-name (match-string 1 p-name))
+
+        when (string-match "^'\\(.*\\)'$" p-name)
+        do (setq p-name (match-string 1 p-name))
+
+        until (string= start-of-loop p-name)))
 
       ;; If the address is 'foo@bar.com <foo@bar.com>' then show just
       ;; 'foo@bar.com'.
 
                  "foo (at home) <foo@bar.com>"
                  "foo [at home] <foo@bar.com>"
                  "Foo Bar"
+                 "'Foo Bar' <foo@bar.com>"
+                 "\"'Foo Bar'\" <foo@bar.com>"
+                 "'\"Foo Bar\"' <foo@bar.com>"
+                 "'\"'Foo Bar'\"' <foo@bar.com>"
                  "Fred Dibna \\[extraordinaire\\] <fred@dibna.com>"))
         (expected '("ДБ <db-uknot@stop.me.uk>"
                     "foo (at home) <foo@bar.com>"
                     "foo [at home] <foo@bar.com>"
                     "Foo Bar"
+                    "Foo Bar <foo@bar.com>"
+                    "Foo Bar <foo@bar.com>"
+                    "Foo Bar <foo@bar.com>"
+                    "Foo Bar <foo@bar.com>"
                     "Fred Dibna [extraordinaire] <fred@dibna.com>"))
         (output (mapcar #'notmuch-show-clean-address input)))
     (notmuch-test-expect-equal output expected)))