I've done some more tests and the set-cookie
header is omitted only on the Windows Phone Emulator. When debugging with an actual device, the header is received as expected.
It is still pretty strange to me why the emulator behaves this way. I saw many posts on issues with http-only cookies in the emulator but none with a concrete reason.
UPDATE:
Testing on the 8.0.10322 emulator works just fine - cookies are handled correctly.It looks as the default phone emulator does something fishy with the cookies.