The WebBrowser control is not a silverlight object and sits on top of all content. You need to flip Visibility between the WebBrowser control and a rectangle with the fill set to WebBrowserBrush when the combobox dropdown is visible and back when it's not. something like:
<WebBrowser x:Name="browserControl" Source="{Binding URL}" />
<Rectangle x:Name="browserControlRect" >
<Rectangle.Fill>
<WebBrowserBrush SourceName="browserControl" />
</Rectangle.Fill>
</Rectangle>
Then either VSM, Behaviours, or code to do the toggling.