my $fileHandle;  #Closes for you when it goes out of scope!
open $fileHandle, '>', $fileName or die "Can't open $fileName: $!\n";
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Good suggestion (though to parallel my original one, I would have to use our $fileHandle instead of my). Actually, although it does not directly answer my original question (where I was looking for a solution using a global file handle, i.e. one written without a sigil), I think it complements nicely my "list of alternative solutions" which I had compiled in my posting.