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In this paper we prove one side divisibility of the Iwasawa-Greenberg mainconjecture for Rankin-Selberg product of a weight two cusp form and an ordinaryCM form of higher weight, using congruences between Klingen Eisenstein seriesand cusp forms on $\mathrm{GU}(3,1)$, generalizing earlier result of thethird-named author to allow non-ordinary cusp forms. The main result is a keyinput in the third author's proof for Kobayashi's $\pm$-main conjecture forsupersingular elliptic curves. The new ingredient here is developing asemi-ordinary Hida theory along an appropriate smaller weight space, and astudy of the semi-ordinary Eisenstein family.

Iwasawa-Greenberg main conjecture for non-ordinary modular forms and Eisenstein congruences on $\mathrm{GU}(3,1)$