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Singularity or negativity of Glauber P-function is a widespread notion of nonclassicality, with important implications in quantum optics and with the character of an irreducible resource. Here we explore how P-nonclassicality may be generated by conditional Gaussian measurements on bipartite Gaussian states. This nonclassical steering may occur in a weak form, which does not imply entanglement, and in a strong form that implies EPR-steerability and thus entanglement. We show that field quadratures are the best measurements to remotely generate nonclassicality, and exploit this result to derive necessary and sufficient conditions for weak and strong nonclassical steering. For two-mode squeezed thermal states (TMST), weak and strong nonclassical steering coincide, and merge with the notion of EPR steering. This also provides a new operational interpretation for P-function nonclassicality as the distinctive feature that allows one-party entanglement verification on TMSTs.