As far as I know Additive Secret Sharing uses a finite field to generate its shares. The gist of the scheme is that the shares $A_{shares}$ = {$A_1, A_2, ..., A_n$} of a value $A$ on a finite field $N$ can be attained by:
$A\ mod\ N = \sum_{A_i \in A_{shares}} (A_i\ mod\ N)$
If all shares in $A_{shares}$ are positive and no finite field was used, an adversary could infer from any given share that the target $A$ is greater than the observed share. The use of the field stops the adversary from learning that.
Alternatively, I believe we could also use negative shares to compensate. If the shares can be negative, others could be greater than the original value and the adversary would learn nothing from them.
I tested this by sampling shares using a uniform distribution over a large range of positive and negative numbers. I found no bias in the results or clues about the value of $A$ from summing and averaging random shares, however I could be missing something.
Everywhere I searched the use of a finite field was used; however I don't understand if it is mandatory (and if so, why).
My question is: Is the use negative valued shares viable as an alternative to computing the Additive Secret Sharing scheme over a finite field, or does it introduce some exploitable bias and is not secure?