Consider $n$ independent and identically distributed (i.i.d.) random variables $X_1, \ldots,X_n$, with (absolutely) continuous distribution $F$, whose (Lebesgue) density is denoted by $f$. Denote by $X_{(1)}<X_{(2)}<\ldots < X_{(n)}$ the order statistics of the sample. Then, it is well known that the joint probability density of $(X_{(1)},X_{(2)},\ldots , X_{(n)})$ is $$ n!\prod_{i=1}^n f(x_i), \quad x_1<x_2<\ldots<x_n. $$
Now, consider $n$ i.i.d. random vectors $(X_1,Y_1), \ldots,(X_n,Y_n)$ with common (absolutely) continuous distribution $G$, whose bivariate (Lebesgue) density is denoted by $g$. Let $(X_{(1)},Y_{(1)}), \ldots,(X_{(n)},Y_{(n)})$ correspond to a reordering of $(X_i,Y_i)$, $1 \leq i \leq n$, based only on the order statistics of the first component, i.e. $$ X_{(1)}<X_{(2)}<\ldots<X_{(n)} $$ but, for $i <j$, not necessarily $Y_{(i)}<Y_{(j)}$, more precisely:
Given a realization $(x_1,y_1),\ldots,(x_n,y_n)$, if for some $l \in \{0,1,\ldots,n-1\}$ the realization $x_i$ corresponds to the $l+1$-th largest value among those observed for the first component, then we simply have $(x_{(n-l)},y_{(n-l)})=(x_i,y_i)$.
QUESTION Which is the joint density of $(X_{(1)},Y_{(1)}), \ldots,(X_{(n)},Y_{(n)})$?