Fix: Delay showing tooltip during page transition (#167614)

Fix: Delay showing tooltip during page transition
fixes: #167359 

## Pre-launch Checklist

- [x] I read the [Contributor Guide] and followed the process outlined
there for submitting PRs.
- [x] I read the [Tree Hygiene] wiki page, which explains my
responsibilities.
- [x] I read and followed the [Flutter Style Guide], including [Features
we expect every widget to implement].
- [x] I signed the [CLA].
- [x] I listed at least one issue that this PR fixes in the description
above.
- [x] I updated/added relevant documentation (doc comments with `///`).
- [x] I added new tests to check the change I am making, or this PR is
[test-exempt].
- [x] I followed the [breaking change policy] and added [Data Driven
Fixes] where supported.
- [x] All existing and new tests are passing.

packages/flutter/lib/src/material/tooltip.dart

@@ -564,6 +564,7 @@ class TooltipState extends State<Tooltip> with SingleTickerProviderStateMixin {
       if (!_visible) {
         return;
       }
+
       _controller.forward();
       _timer?.cancel();
       _timer = showDuration == null ? null : Timer(showDuration, _controller.reverse);
@@ -927,8 +928,16 @@ class TooltipState extends State<Tooltip> with SingleTickerProviderStateMixin {
       child: widget.child,
     );
 
+    bool visible = _visible;
+
+    // Check if there's an ongoing route transition
+    final ModalRoute<dynamic>? route = ModalRoute.of(context);
+    if (route?.secondaryAnimation != null && route!.secondaryAnimation!.isAnimating) {
+      visible = false;
+    }
+
     // Only check for gestures if tooltip should be visible.
-    if (_visible) {
+    if (visible) {
       result = _ExclusiveMouseRegion(
         onEnter: _handleMouseEnter,
         onExit: _handleMouseExit,

packages/flutter/test/material/tooltip_test.dart

@@ -4,6 +4,7 @@
 
 import 'dart:ui';
 
+import 'package:flutter/cupertino.dart';
 import 'package:flutter/foundation.dart';
 import 'package:flutter/gestures.dart';
 import 'package:flutter/material.dart';
@@ -3478,6 +3479,49 @@ void main() {
     );
     expect(tester.element(textAncestors.first).size, equals(tooltipConstraints.biggest));
   });
+
+  // This is a regression test for https://github.com/flutter/flutter/issues/167359.
+  testWidgets('Tooltip does not show while transitioning from another page', (
+    WidgetTester tester,
+  ) async {
+    await tester.pumpWidget(
+      MaterialApp(
+        home: Scaffold(
+          appBar: AppBar(
+            leading: const Center(child: Tooltip(message: 'Hello', child: Text('World'))),
+          ),
+          body: Builder(
+            builder: (BuildContext context) {
+              return TextButton(
+                onPressed:
+                    () => Navigator.push(
+                      context,
+                      CupertinoPageRoute<void>(
+                        builder:
+                            (BuildContext context) =>
+                                Scaffold(appBar: AppBar(title: const Text('Second Page'))),
+                      ),
+                    ),
+                child: const Text('Go to Second Page'),
+              );
+            },
+          ),
+        ),
+      ),
+    );
+
+    await tester.tap(find.text('Go to Second Page'));
+    await tester.pumpAndSettle();
+
+    final TestGesture gesture = await tester.createGesture(kind: PointerDeviceKind.mouse);
+    await gesture.addPointer();
+    await tester.tap(find.byType(BackButton));
+    await tester.pump(const Duration(milliseconds: 250));
+    await gesture.moveTo(tester.getCenter(find.text('World')));
+    await tester.pumpAndSettle();
+
+    expect(tester.takeException(), isNull);
+  });
 }
 
 Future<void> setWidgetForTooltipMode(