  Nonclassicality according to the singularity or negativity of the Glauber
P-function is a powerful resource in quantum information, with relevant
implications in quantum optics. In a Gaussian setting, and for a system of two
modes, we explore how P-nonclassicality may be conditionally generated or
influenced on one mode by Gaussian measurements on the other mode. Starting
from the class of two-mode squeezed thermal states (TMST), we introduce the
notion of nonclassical steering (NS) and the graphical tool of Gaussian
triangoloids. In particular, we derive a necessary and sufficient condition for
a TMST to be nonclassically steerable, and show that entanglement is only
necessary. We also apply our criterion to noisy propagation of a twin-beam
state, and evaluate the time after which NS is no longer achievable. We then
generalize the notion of NS to the full set of Gaussian states of two modes,
and recognize that it may occur in a weak form, which does not imply
entanglement, and in a strong form that implies EPR-steerability and, a
fortiori, also entanglement. These two types of NS coincide exactly for TMSTs,
and they merge with the previously known notion of EPR steering. By the same
token, we recognize a new operational interpretation of P-nonclassicality: it
is the distinctive property that allows one-party entanglement verification on
TMSTs.
