I found an answer here
To fix that but we have to create a new autocompletebox and override the OnApplyTemplate method.
public class CustomAutoComplete : AutoCompleteBox
{
TextBox mytext;
public override void OnApplyTemplate()
{
base.OnApplyTemplate();
mytext = GetTemplateChild("Text") as TextBox;
mytext.TextChanged += new System.Windows.Controls.TextChangedEventHandler(mytext_TextChanged);
}
void mytext_TextChanged(object sender, System.Windows.Controls.TextChangedEventArgs e)
{
this.Text = mytext.Text;
OnTextChanged(new RoutedEventArgs());
}
}