jmc

We will begin by finding all the positive factors of $-12$, which are the same as the positive factors of $12$. The positive factors of 12 are 1, 2, 3, 4, 6, and 12. The four numbers we seek must be among these six numbers.

Note that the number $4$ is not a factor of each number on the list, since dividing $114$ by $4$ gives a remainder of $2$. We also know that $12$ cannot be a factor of $114$, since dividing $114$ by $12$ gives a remainder of $6$. However, $6$ is a factor of each number on the list, since $$\begin{array}{rl}36& =6\cdot 6\\ 72& =6\cdot 12\\ -12& =6\cdot (-2)\\ 96& =6\cdot 16\\ 114& =6\cdot 19\end{array}$$Since $1$, $2$, $3$, and $6$ are factors of $6$, and $6$ is a factor of each number on the list, $1$, $2$, $3$, and $6$ must be a factor of each number on the list. So these are the four numbers we were looking for, and our final answer is $$1+2+3+6=\boxed{12}.$$