It turns out it is super-easy to use the Sort(Delegate) method to create your own sorting method and then pass it to the call to sort.
The sorting method would use StrComp to get a case-sensitive sort and could look like this:
Public Function CaseSensitiveSort(value1 As String, value2 As String) as Integer Return StrComp(value1, value2, Realbasic.StrCompCaseSensitive) End Function
And then you can call it like this:
Dim s() As String = Array("alpha", "Test", "Beta", "candy") s.Sort(AddressOf CaseSensitiveSort)
To get this sorted result: