Without an agreement to accept each other's paper, you are on your own. There is more than one form of interline agreements - the MITA (IATA's Multilateral Interline Agreement) or Bilateral agreements. Agreements can be 'one way' as well. i.e. One will accept the other's tickets but not vice-versa.
I found connections for FAE to KTT but that involved Atlantic/SK/AY. As long as the ticket issuing airline has agreements with all accepting airlines it is valid even if the accepting airlines do not agreements with each other. In this case, for example, Finnair may not have an agreement with Atlantic (and vv) but a through ticket can still be issued on SK as it has agreements with the other two.
I fear that MITA is going the way of the Dodo, to be replaced by exclusive alliance agreements. Life is going to become more difficult.