  Given a presilting object in a triangulated category, we find necessary and
sufficient conditions for the existence of a complement. This is done both for
classic (pre)silting objects and for large (pre)silting objects. The key
technique is the study of associated co-t-structures. As a consequence of our
techniques we recover some known cases of the existence of complements,
including for derived categories of some hereditary abelian categories and for
silting-discrete algebras. Moreover, we also show that a finite-dimensional
algebra is silting discrete if and only if every bounded large silting complex
is equivalent to a compact one.
