WPF DataGrid: Automatically re-sort on a DataGridTemplateColumn

Question

In WPF's DataGrid control, if you set a column to one of the default column types (like DataGridTextColumn or DataGridCheckBoxColumn), sort on that column, and then change its value, the grid will automatically be re-sorted.

However, if you use a DataGridTemplateColumn (and allow the column to be sorted), it can be sorted, but changing the value of a cell in this column does not cause the grid is not re-sorted. How can I coax it into automatically triggering a re-sort?

XAML:

<DataGrid Name="grid" AutoGenerateColumns="False">
  <DataGrid.Columns>
    <DataGridTextColumn Header="First name" Binding="{Binding First}"/>
    <DataGridTemplateColumn Header="Last name" SortMemberPath="Last">
      <DataGridTemplateColumn.CellTemplate>
        <DataTemplate>
          <TextBox Text="{Binding Last}"/>
        </DataTemplate>
      </DataGridTemplateColumn.CellTemplate>
    </DataGridTemplateColumn>
  </DataGrid.Columns>
</DataGrid>

Binding:

ObservableCollection items = new ObservableCollection();
grid.ItemsSource = items;
items.Add(new Character() { First = "Homer", Last = "Simpson" });
items.Add(new Character() { First = "Kent", Last = "Brockman" });
items.Add(new Character() { First = "Montgomery", Last = "Burns" });

Here's my item class, just in case that's relevant:

public class Character : INotifyPropertyChanged {
    private string first, last;
    public event PropertyChangedEventHandler PropertyChanged;
    private void Notify(string name) {
        if (PropertyChanged != null)
            PropertyChanged(this, new PropertyChangedEventArgs(name));
    }
    public string First { get { return first; } set { first = value; Notify("First"); } }
    public string Last { get { return last; } set { last = value; Notify("Last"); } }
}