  We introduce a new feature of no-signaling (Bell) non-local theories, namely,
when a system of multiple parties manifests genuine non-local correlation, then
there cannot be arbitrarily high non-local correlation among any subset of the
parties. We call this feature, \textit{complementarity of genuine multipartite
non-locality}. We use Svetlichny's criterion for genuine multipartite
non-locality and non-local games to derive the complementarity relations under
no-signaling constraints. We find that the complementarity relations are
tightened for the much stricter quantum constraints. We compare this notion
with the well-known notion of \textit{monogamy of non-locality}. As a
consequence, we obtain tighter non-trivial monogamy relations that take into
account genuine multipartite non-locality. Furthermore, we provide numerical
evidence showcasing this feature using a bipartite measure and several other
well-known tripartite measures of non-locality.
