  We show that certain $p$-adic Eisenstein series for quaternionic modular
groups of degree 2 become "real" modular forms of level $p$ in almost all
cases. To prove this, we introduce a $U(p)$ type operator. We also show that
there exists a $p$-adic Eisenstein series of the above type that has
transcendental coefficients. Former examples of $p$-adic Eisenstein series for
Siegel and Hermitian modular groups are both rational (i.e., algebraic).
