TextEditingController makes widget lose its previous state

The explanation for the observed behavior is the following:

CupertinoTextField uses an internal TextEditingController for which the framework automatically sets an AutomaticKeepAlive. This keepAlive is responsible for keeping the state.

If you use your own controller, you are in charge of attaching the AutomaticKeepAlive because the framework doesn't do it for you.

The following snippet adds the keepAlive to your code:

import 'package:flutter/cupertino.dart';
import 'package:flutter/material.dart';

void main() => runApp(new MyApp());

class MyApp extends StatelessWidget {
  @override
  Widget build(BuildContext context) {
    return MaterialApp(
      title: 'test',
      home: DefaultTabController(
        length: 2,
        child: Scaffold(
            body: TabBarView(
              children: [
                new Search(),
                new Setting(),
              ],
            ),
            bottomNavigationBar: Container(
              height: 60,
              child: new TabBar(
                tabs: [
                  Tab(icon: new Icon(Icons.search)),
                  Tab(icon: new Icon(Icons.settings)),
                ],
                labelColor: Colors.blue,
                unselectedLabelColor: Colors.grey,
              ),
            )),
      ),
    );
  }
}

class Setting extends StatelessWidget {
  @override
  Widget build(BuildContext context) {
    return IconButton(
        icon: Icon(Icons.check),
        onPressed: () {
          Navigator.push(
              context,
              CupertinoPageRoute(
                  builder: (context) => new Scaffold(
                        appBar: AppBar(
                          title: Text('3'),
                        ),
                      )));
        });
  }
}

class Search extends StatefulWidget {
  @override
  createState() => new SearchState();
}

class SearchState extends State<Search> with AutomaticKeepAliveClientMixin {
  String currentWord = '';
  final TextEditingController textController = new TextEditingController();

  @override
  void initState() {
    super.initState();
    textController?.addListener(updateKeepAlive);
  }

  @override
  void dispose() {
    textController?.removeListener(updateKeepAlive);
    textController.dispose();
    super.dispose();
  }

  @override
  Widget build(BuildContext context) {
    super.build(context); // See AutomaticKeepAliveClientMixin.
    return new Scaffold(
        appBar: new AppBar(
          title: new Row(
            children: <Widget>[
              new Expanded(
                child: new CupertinoTextField(
                  style: TextStyle(color: Colors.white),
                  cursorColor: Colors.white,
                  controller: textController,
                  maxLines: 1,
                  clearButtonMode: OverlayVisibilityMode.editing,
                  onChanged: (text) {
                    setState(() {
                      currentWord = text;
                    });
                  },
                ),
              ),
            ],
          ),
        ),
        body: ListView.builder(
            itemCount: 5,
            itemBuilder: (context, i) {
              return Text(currentWord);
            }));
  }

  @override
  bool get wantKeepAlive => textController?.text?.isNotEmpty == true;
}