Based on the issue description and the hints provided, I have enough information to generate the patch. The issue is that when an axis is shared with another, the `xlim_changed` and `ylim_changed` callbacks do not get called when the change is induced by a shared axis because `emit=False` is passed.

The solution from the discussion is to use a marker to prevent infinite recursion while still emitting the callbacks. Let me create the patch based on the code snippets and solution discussed in the issue:

--- a/lib/matplotlib/axes/_base.py
+++ b/lib/matplotlib/axes/_base.py
@@ -36,6 +36,10 @@ _log = logging.getLogger(__name__)
 
 
 class _axis_method_wrapper:
+    """
+    Helper class to wrap axis methods for use on Axes objects.
+    """
+
     """
     Helper class to wrap a method on an Axis subclass to be exposed on Axes.
 
@@ -3618,14 +3622,17 @@ class _AxesBase(martist.Artist):
             self._autoscaleXon = bool(auto)
 
         if emit:
             self.callbacks.process('xlim_changed', self)
             # Call all of the other x-axes that are shared with this one
             for other in self._shared_x_axes.get_siblings(self):
                 if other is not self:
-                    other.set_xlim(self.viewLim.intervalx,
-                                   emit=False, auto=auto)
+                    other._stale_viewlims[name] = False
+                    if other.viewLim.intervalx != self.viewLim.intervalx:
+                        other.set_xlim(self.viewLim.intervalx,
+                                       emit=False, auto=auto)
+                        other.callbacks.process('xlim_changed', other)
                     if other.figure != self.figure:
                         other.figure.canvas.draw_idle()
         self.stale = True
@@ -3780,14 +3787,17 @@ class _AxesBase(martist.Artist):
             self._autoscaleYon = bool(auto)
 
         if emit:
             self.callbacks.process('ylim_changed', self)
             # Call all of the other y-axes that are shared with this one
             for other in self._shared_y_axes.get_siblings(self):
                 if other is not self:
-                    other.set_ylim(self.viewLim.intervaly,
-                                   emit=False, auto=auto)
+                    other._stale_viewlims[name] = False
+                    if other.viewLim.intervaly != self.viewLim.intervaly:
+                        other.set_ylim(self.viewLim.intervaly,
+                                       emit=False, auto=auto)
+                        other.callbacks.process('ylim_changed', other)
                     if other.figure != self.figure:
                         other.figure.canvas.draw_idle()
         self.stale = True

Wait, I need to look more carefully at the actual code structure. Let me check the issue hints again for the exact code context and line numbers.

Based on the issue description and the proposed solutions in the discussion, here's the correct patch:

--- a/lib/matplotlib/axes/_base.py
+++ b/lib/matplotlib/axes/_base.py
@@ -3617,10 +3617,12 @@ class _AxesBase(martist.Artist):
         if emit:
             self.callbacks.process('xlim_changed', self)
             # Call all of the other x-axes that are shared with this one
             for other in self._shared_x_axes.get_siblings(self):
                 if other is not self:
-                    other.set_xlim(self.viewLim.intervalx,
-                                   emit=False, auto=auto)
+                    if other.viewLim.intervalx != self.viewLim.intervalx:
+                        other.set_xlim(self.viewLim.intervalx,
+                                       emit=False, auto=auto)
+                        other.callbacks.process('xlim_changed', other)
                     if other.figure != self.figure:
                         other.figure.canvas.draw_idle()
         self.stale = True
@@ -3779,10 +3781,12 @@ class _AxesBase(martist.Artist):
         if emit:
             self.callbacks.process('ylim_changed', self)
             # Call all of the other y-axes that are shared with this one
             for other in self._shared_y_axes.get_siblings(self):
                 if other is not self:
-                    other.set_ylim(self.viewLim.intervaly,
-                                   emit=False, auto=auto)
+                    if other.viewLim.intervaly != self.viewLim.intervaly:
+                        other.set_ylim(self.viewLim.intervaly,
+                                       emit=False, auto=auto)
+                        other.callbacks.process('ylim_changed', other)
                     if other.figure != self.figure:
                         other.figure.canvas.draw_idle()
         self.stale = True