How to use a Positioned widget in an AppBar

From what I've read this should be possible, but I can't quite get it to work. I have a Stack inside the bottom of an appBar, there's a Positioned list inside of the Stack. Everything seems to be positioned as expected, but the appBar is cropping the list, so the list isn't displaying on top if the appBar and contents of the body.

I'm new to Flutter, but in the world of HTML I'd have an absolutely positioned list and the appBar would be fixed with a z-index higher than the body allowing for the layered effect.

I've tried a bunch of variations, but it just seems the appBar wants to crop it's children. Any help would be appreciated.

Here's a pic of what I'm trying to emulate:

Here's a snippet of code:

return new Scaffold(
  appBar: new AppBar(
    title: new Row(
      children: <Widget>[
        new Padding(
          padding: new EdgeInsets.only(
            right: 10.0,
          ),
          child: new Icon(Icons.shopping_basket),
        ),
        new Text(appTitle)
      ],
    ),
    bottom: new PreferredSize(
      preferredSize: const Size.fromHeight(30.0),
      child: new Padding(
        padding: new EdgeInsets.only(
          bottom: 10.0,
          left: 10.0,
          right: 10.0,
        ),
        child: new AutoCompleteInput(
          key: new ObjectKey('$completionList'),
          completionList: completionList,
          hintText: 'Add Item',
          onSubmit: _addListItem,
        ),
      ),
    ),
  ),

Update #1

Widget build(BuildContext ctx) {
  final OverlayEntry _entry = new OverlayEntry(
    builder: (BuildContext context) => const Text('hi')
  );
  Overlay.of(ctx, debugRequiredFor: widget).insert(_entry);
  return new Row(

Created an example file that demonstrates what I came up with (at least what's related to this question). Hopefully it saves others from any unnecessary headaches.

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

String appTitle = 'Overlay Example';

class _CustomDelegate extends SingleChildLayoutDelegate {
  final Offset target;
  final double verticalOffset;

  _CustomDelegate({
    @required this.target,
    @required this.verticalOffset,
  }) : assert(target != null),
       assert(verticalOffset != null);

  @override
  BoxConstraints getConstraintsForChild(BoxConstraints constraints) => constraints.loosen();

  @override
  Offset getPositionForChild(Size size, Size childSize) {
    return positionDependentBox(
      size: size,
      childSize: childSize,
      target: target,
      verticalOffset: verticalOffset,
      preferBelow: true,
    );
  }

  @override
  bool shouldRelayout(_CustomDelegate oldDelegate) {
    return
      target != oldDelegate.target
      || verticalOffset != oldDelegate.verticalOffset;
  }
}

class _CustomOverlay extends StatelessWidget {
  final Widget child;
  final Offset target;

  const _CustomOverlay({
    Key key,
    this.child,
    this.target,
  }) : super(key: key);

  @override
  Widget build(BuildContext context) {
    double borderWidth = 2.0;
    Color borderColor = Theme.of(context).accentColor;

    return new Positioned.fill(
      child: new IgnorePointer(
        ignoring: false,
        child: new CustomSingleChildLayout(
          delegate: new _CustomDelegate(
            target: target,
            verticalOffset: -5.0,
          ),
          child: new Padding(
            padding: const EdgeInsets.symmetric(horizontal: 10.0),
            child: new ConstrainedBox(
              constraints: new BoxConstraints(
                maxHeight: 100.0,
              ),
              child: new Container(
                decoration: new BoxDecoration(
                  color: Colors.white,
                  border: new Border(
                    right: new BorderSide(color: borderColor, width: borderWidth),
                    bottom: new BorderSide(color: borderColor, width: borderWidth),
                    left: new BorderSide(color: borderColor, width: borderWidth),
                  ),
                ),
                child: child,
              ),
            ),
          ),
        ),
      ),
    );
  }
}

class _CustomInputState extends State<_CustomInput> {
  TextEditingController _inputController = new TextEditingController();
  FocusNode _focus = new FocusNode();
  List<String> _listItems;
  OverlayState _overlay;
  OverlayEntry _entry;
  bool _entryIsVisible = false;
  StreamSubscription _sub;

  void _toggleEntry(show) {
    if(_overlay.mounted && _entry != null){
      if(show){
        _overlay.insert(_entry);
        _entryIsVisible = true;
      }
      else{
        _entry.remove();
        _entryIsVisible = false;
      }
    }
    else {
      _entryIsVisible = false;
    }
  }

  void _handleFocus(){
    if(_focus.hasFocus){
      _inputController.addListener(_handleInput);
      print('Added input handler');
      _handleInput();
    }
    else{
      _inputController.removeListener(_handleInput);
      print('Removed input handler');
    }
  }

  void _handleInput() {
    String newVal = _inputController.text;

    if(widget.parentStream != null && _sub == null){
      _sub = widget.parentStream.listen(_handleStream);
      print('Added stream listener');
    }

    if(_overlay == null){
      final RenderBox bounds = context.findRenderObject();
      final Offset target = bounds.localToGlobal(bounds.size.bottomCenter(Offset.zero));

      _entry = new OverlayEntry(builder: (BuildContext context){
        return new _CustomOverlay(
          target: target,
          child: new Material(
            child: new ListView.builder(
              padding: const EdgeInsets.all(0.0),
              itemBuilder: (BuildContext context, int ndx) {
                String label = _listItems[ndx];
                return new ListTile(
                  title: new Text(label),
                  onTap: () {
                    print('Chose: $label');
                    _handleSubmit(label);
                  },
                );
              },
              itemCount: _listItems.length,
            ),
          ),
        );
      });
      _overlay = Overlay.of(context, debugRequiredFor: widget);
    }

    setState(() {
      // This can be used if the listItems get updated, which won't happen in
      // this example, but I figured it was useful info.
      if(!_entryIsVisible && _listItems.length > 0){
        _toggleEntry(true);
      }else if(_entryIsVisible && _listItems.length == 0){
        _toggleEntry(false);
      }else{
        _entry.markNeedsBuild();
      }
    });
  }

  void _exitInput(){
    if(_sub != null){
      _sub.cancel();
      _sub = null;
      print('Removed stream listener');
    }
    // Blur the input
    FocusScope.of(context).requestFocus(new FocusNode());
    // hide the list
    _toggleEntry(false);

  }

  void _handleSubmit(newVal) {
    // Set to selected value
    _inputController.text = newVal;
    _exitInput();
  }

  void _handleStream(ev) {
    print('Input Stream : $ev');
    switch(ev){
      case 'TAP_UP':
        _exitInput();
        break;
    }
  }

  @override
  void initState() {
    super.initState();
    _focus.addListener(_handleFocus);
    _listItems = widget.listItems;
  }

  @override
  void dispose() {
    _inputController.removeListener(_handleInput);
    _inputController.dispose();

    if(mounted){
      if(_sub != null) _sub.cancel();
      if(_entryIsVisible){
        _entry.remove();
        _entryIsVisible = false;
      }
      if(_overlay != null && _overlay.mounted) _overlay.dispose();
    }

    super.dispose();
  }

  @override
  Widget build(BuildContext ctx) {
    return new Row(
      children: <Widget>[
        new Expanded(
          child: new TextField(
            autocorrect: true,
            focusNode: _focus,
            controller: _inputController,
            decoration: new InputDecoration(
              border: new OutlineInputBorder(
                borderRadius: const BorderRadius.all(
                  const Radius.circular(5.0),
                ),
                borderSide: new BorderSide(
                  color: Colors.black,
                  width: 1.0,
                ),
              ),
              contentPadding: new EdgeInsets.all(10.0),
              filled: true,
              fillColor: Colors.white,
            ),
            onSubmitted: _handleSubmit,
          ),
        ),
      ]
    );
  }
}

class _CustomInput extends StatefulWidget {
  final List<String> listItems;
  final Stream parentStream;

  _CustomInput({
    Key key,
    this.listItems,
    this.parentStream,
  }): super(key: key);

  @override
  State createState() => new _CustomInputState();
}

class HomeState extends State<Home> {
  List<String> _overlayItems = [
    'Item 01',
    'Item 02',
    'Item 03',
  ];
  StreamController _eventDispatcher = new StreamController.broadcast();

  Stream get _stream => _eventDispatcher.stream;

  _onTapUp(TapUpDetails details) {
    _eventDispatcher.add('TAP_UP');
  }

  @override
  void initState() {
    super.initState();
  }

  @override
  void dispose(){
    super.dispose();
    _eventDispatcher.close();
  }

  @override
  Widget build(BuildContext context){
    return new GestureDetector(
      onTapUp: _onTapUp,
      child: new Scaffold(
        appBar: new AppBar(
          title: new Row(
            children: <Widget>[
              new Padding(
                padding: new EdgeInsets.only(
                  right: 10.0,
                ),
                child: new Icon(Icons.layers),
              ),
              new Text(appTitle)
            ],
          ),
          bottom: new PreferredSize(
            preferredSize: const Size.fromHeight(30.0),
            child: new Padding(
              padding: new EdgeInsets.only(
                bottom: 10.0,
                left: 10.0,
                right: 10.0,
              ),
              child: new _CustomInput(
                key: new ObjectKey('$_overlayItems'),
                listItems: _overlayItems,
                parentStream: _stream,
              ),
            ),
          ),
        ),
        body: const Text('Body content'),
      ),
    );
  }
}

class Home extends StatefulWidget {
  @override
  State createState() => new HomeState();
}

void main() => runApp(new MaterialApp(
  title: appTitle,
  home: new Home(),
));