  Employing a procedure called monitoring---via a completely positive
trace-preserving map that is able to interpolate between weak and projective
measurements---we investigate the resilience of the recently proposed
realism-based nonlocality to local and bilocal weak measurements. This analysis
indicates realism-based nonlocality as the most ubiquitous and persistent form
of quantumness within a wide class of quantum-correlation quantifiers. In
particular, we show that the set of states possessing this type of quantumness
forms a strict superset of symmetrically discordant states and, therefore, of
discordant, entangled, steerable, and Bell-nonlocal states. Moreover, we find
that, under monitoring, realism-based nonlocality is not susceptible to sudden
death.
