A transparent Border was added as an invisible overlay on top of the Run element within a Grid. This Border allows tooltips to be displayed without altering the Run element's behavior or appearance. The tooltip properties are be applied to the Border, ensuring the tooltip functionality works as expected. Below is the change made:

<ControlTemplate x:Key="CustomTemplate" TargetType="TextBox">
<Grid>     
<TextBlock FontSize="{TemplateBinding FontSize}" TextWrapping="Wrap" TextTrimming="CharacterEllipsis"                AutomationProperties.AutomationId="CustomTextBlockId">         
<TextBlock Style="{StaticResource CustomStyle}" TextWrapping="Wrap" Text="{Binding Path=DisplayText}" />     
</TextBlock>
<Border Background="Transparent"
    ToolTipService.ToolTipType="Auto"
    ToolTipService.ToolTipHeader="{Binding Path=ToolTip.ToolTipHeader}"
    ToolTipService.ToolTipBody="{Binding Path=ToolTip.ToolTipBody}">
</Border>
</Grid>
</ControlTemplate>