diff --git a/lisp/international/mule-cmds.el b/lisp/international/mule-cmds.el
index d361971a1f..d59f2c0ebf 100644
--- a/lisp/international/mule-cmds.el
+++ b/lisp/international/mule-cmds.el
@@ -3084,13 +3084,11 @@ ucs-names
         (puthash "BELL (BEL)" ?\a names)
         (setq ucs-names names))))
 
-(defun mule--ucs-names-annotation (name)
-  ;; FIXME: It would be much better to add this annotation before rather than
-  ;; after the char name, so the annotations are aligned.
-  ;; FIXME: The default behavior of displaying annotations in italics
-  ;; doesn't work well here.
-  (let ((char (gethash name ucs-names)))
-    (when char (format " (%c)" char))))
+(defun mule--ucs-names-affixation (names)
+  (mapcar (lambda (name)
+            (let ((char (gethash name ucs-names)))
+              (list name (concat (if char (format "%c" char) " ") "\t") "")))
+          names))
 
 (defun char-from-name (string &optional ignore-case)
   "Return a character as a number from its Unicode name STRING.
@@ -3133,13 +3131,14 @@ read-char-by-name
 as names, not numbers."
   (let* ((enable-recursive-minibuffers t)
 	 (completion-ignore-case t)
+	 (completion-tab-width 4)
 	 (input
 	  (completing-read
 	   prompt
 	   (lambda (string pred action)
 	     (if (eq action 'metadata)
 		 '(metadata
-		   (annotation-function . mule--ucs-names-annotation)
+		   (affixation-function . mule--ucs-names-affixation)
 		   (category . unicode-name))
 	       (complete-with-action action (ucs-names) string pred)))))
 	 (char
diff --git a/lisp/minibuffer.el b/lisp/minibuffer.el
index 87bf3d36fa..64498b3729 100644
--- a/lisp/minibuffer.el
+++ b/lisp/minibuffer.el
@@ -1134,6 +1134,7 @@ completion--cycle-threshold
 (defvar-local completion-all-sorted-completions nil)
 (defvar-local completion--all-sorted-completions-location nil)
 (defvar completion-cycling nil)      ;Function that takes down the cycling map.
+(defvar completion-tab-width 8)
 
 (defvar completion-fail-discreetly nil
   "If non-nil, stay quiet when there  is no match.")
@@ -1718,6 +1719,10 @@ completion--insert-strings
 	   (row 0)
            (first t)
 	   (laststring nil))
+      (unless (or tab-stop-list (zerop (mod colwidth completion-tab-width)))
+        ;; Align to tab positions for the case
+        ;; when the caller uses tabs inside prefix.
+        (setq colwidth (- colwidth (mod colwidth completion-tab-width))))
       ;; The insertion should be "sensible" no matter what choices were made
       ;; for the parameters above.
       (dolist (str strings)
@@ -1758,9 +1763,10 @@ completion--insert-strings
 		  ;; already past the goal column, there is still
 		  ;; a space displayed.
 		  (set-text-properties (1- (point)) (point)
-				       ;; We can't just set tab-width, because
-				       ;; completion-setup-function will kill
-				       ;; all local variables :-(
+				       ;; We can set tab-width using
+				       ;; completion-tab-width, but
+				       ;; the caller can prefer using
+				       ;; \t to align prefixes.
 				       `(display (space :align-to ,column)))
 		  nil))))
             (setq first nil)
diff --git a/lisp/simple.el b/lisp/simple.el
index c9f4f2bb44..8cd5de20d7 100644
--- a/lisp/simple.el
+++ b/lisp/simple.el
@@ -8822,6 +8822,8 @@ completion-setup-function
 	     insert-fun))
       (set (make-local-variable 'completion-reference-buffer) mainbuf)
       (if base-dir (setq default-directory base-dir))
+      (when completion-tab-width
+        (setq tab-width completion-tab-width))
       ;; Maybe insert help string.
       (when completion-show-help
 	(goto-char (point-min))