I am familiar with the theory of Hida families of modular forms, so Hida theory on $GL_2$, but I am not familiar with Hida theory on any other algebraic groups. My question concerns Hida families of Eisenstein series on $GU(2,2)$.

In Skinner and Urban's paper, *The Iwasawa Main Conjectures for $GL_2$*, they consider Hida families of Eisenstein series on $GU(2,2)$ associated to the data of a representation $\rho_f$ that comes from an ordinary cuspidal eigenform $f$ on $GL_2$, and $\chi$, $\psi$ certain Hecke characters of an imaginary quadratic field $K$ in which $p$ splits. My understanding is that the $p$-adic variation of the Hida family of Eisenstein series on $GU(2,2)$ corresponds to varying $f$ through the Hida family of cusp forms on $GL_2$ passing through $f$ and varying $\chi$ and $\psi$ each through one dimensional $p$-adic families of Hecke characters of $K$.

In Skinner and Urban's paper and everyone else I have looked concerning Hida families of Eisenstein series on $GU(2,2)$, it is assumed that $p$ splits in the imaginary quadratic field $K$. My question is, can one make the same construction and consider a Hida family of Eisenstein series on $GU(2,2)$ associated to the same data except that $p$ is inert in the imaginary quadratic field $K$?

It seems to me that the answer to his question will be no, but I have no idea why. If the answer is no could someone shed some light on why $p$ being split in $K$ is essential or point me to a reference that explains why $p$ must be split?