Usually "exchangeable normal random variables" means jointly normal random variables $X_1,\ldots,X_n$ (i.e. so distributed that every linear combination of them is normally distributed) that are exchangeable in the sense that if no matter you permute the indices, you don't alter the probability distribution of the $n$-tuple.

But I wonder about a different sort of exchangeable normals. Is there some probability distribution of an infinite sequence $X_0,X_1,X_2,\ldots$ of random variables such that

$\bullet$ For each $i$, $X_i \sim N(0,1)$ (marginal normality),

$\bullet$ Finite permutations of the indices never alter the probability distribution of the sequence as a whole (exchangeability),

$\bullet$ Any Borel set $A$, $\lim\limits_{n\to\infty}\Pr(X_0\in A \mid X_1,\ldots,X_n) = \lim\limits_{n\to\infty} \dfrac{|A\cap \lbrace X_1,\ldots,X_n \rbrace|}{n}$

$\bullet$ Any reasonably well behaved distribution could be the limiting distribution; which one it is would determine the nature of the dependence among $X_1,X_2,X_3,\ldots.$

?

(If I'm not mistaken, it would be enough to show the third bullet point is satisfied whenever $A$ is a half-infinite interval.)