Workaround for accessing some ASMX services from Silverlight 4

On this blog we normally talk about building and accessing WCF services, since that is the recommended way to build services for Silverlight. However, we continue to support ASMX services using the familiar "Add Service Reference" feature.
Just recently some folks trying to talk to the ASMX services that SharePoint 2010 exposes brought an interesting issue to my attention. You would hit this issue if you were accessing SharePoint's userprofileservice.asmx, but more generally you could hit this with any ASMX service that uses known types with Guid/char (for example the operation signature returns Object, but you actually return a Guid/char from within the operation).
When you hit this issue, you get the following exception as Silverlight tries to deserialize the service response:
System.ServiceModel.Dispatcher.NetDispatcherFaultException was unhandled by user code. ...