Given an positive integer $n$, we define an order of subset sums (or simply, an order, when there is no ambiguity) to be a sequence of all subsets of $\{1,\ldots,n\}$. For example, when $n=2$, the sequence $\emptyset,\{1\},\{2\},\{1,2\}$ is an order.
We call an order $S_1,\ldots,S_{2^n}$ valid if there exist real numbers $0<x_1<\cdots<x_n$ such that $\sum_{i\in S_1}x_i<\cdots<\sum_{i\in S_{2^n}}x_i$. For example, when $n=2$, the sequence $\emptyset,\{1\},\{2\},\{1,2\}$ is a valid order, but the sequence $\emptyset,\{1\},\{1,2\},\{2\}$ is not because we cannot make $x_1+x_2<x_2$.
The question is, given $n$, how to enumerate all possible valid orders. To output an order $S_1,\ldots,S_{2^n}$, it is sufficient to output an algorithm that generates this order, i.e., an algorithm with no input (the parameter $n$ is built-in) that outputs $S_1,\ldots,S_{2^n}$ in order, as long as the algorithm runs in $O(n2^n)$ time. Even so, this problem still cannot be solved in time polynomial in $n$, because there may be exponentially many valid orders, thus an algorithm running in exponential time is welcome.
A trivial algorithm would be to iterate over all possible orders, and check for each one if it is valid. But I cannot even find an (efficient) way to check if an order is valid.
