The answer is yes. Consider the sequence
$100, 200, 201, 202, 500, 601, 700, 701, 801, 1000, 1194, 1200.$
It is easy to see that $X=\{1,2,5,7,10,12\}$ and $Y=\{3,4,6,8,9,11\}$ are exact. Moreover, we claim that they are the only exact subsets. To see this, note that for every subset $X'$ of $X$, the sum of the corresponding sequence values is divisible by $100$. However, for every proper subset $Y'$ of $Y$, the sum of the corresponding sequence values is not divisible by $100$. Thus, $X$ and $Y$ are the only exact subsets.