Can random variables have the same distribution but different conditional distributions? Can two equally distributed random variables $X$, $Y$, defined over the same probability space have different conditional distributions relative to some $\sigma$-algebra $\mathcal{A}$, so that $X\sim Y$ but $P\left(E\left(\left.f(X)\right|\mathcal{A}\right)\neq E\left(\left.f(Y)\right|\mathcal{A}\right)\right)>0$ for some integrable $f$?

Sure: take some i.i.d. random variables $X$ and $Y$, and $\mathcal A=\sigma(X)$.
