This is something I became aware of while reading a textbook about computer arithmetic (specifically the parts about 2's complement). The book was very non-formal about it, being a hardware textbook, but I conjectured what I think if a reasonable formalization of the idea. For the sake of argument I'll call them hypernatural numbers. I'll stick to the binary version but I'm pretty sure you could define them for an arbitrary base.
Definition: A (base 2) hypernatural number $A$ is a sequence $(A_0,\ A_1,\ ...)$ of binary digits such that for some natural $n$, all $A_i$ with $i \geq n$ are equal. ie, a hypernatural either has infinite leading 1s or infinite leading 0s.
Operations: For the sake of brevity I'll just say: imagine the hypernatural represents a binary number in positional notation, and use the elementary school algorithms to do hypernatural addition and multiplication. You could make this more rigorous by defining a rule to get $(A + B)_i$ and $(A \times B)_i$ as a function of the first $i + 1$ elements of $A$ and $B$. But hopefully you get the idea.
Isomorphism to the relative integers: Basically, the hypernaturals with leading 0s are isomorphic to the natural numbers (obviously), and the hypernaturals with leading 1s are isomorphic to the strictly negative integers. Let the bijection $B$ from leading-0 hypernaturals to naturals mean "interpret the sequence as a binary number". Then $B'(A) = -(B(\bar A) + 1)$ from leading-1 hypernaturals to negative integers, where the overbar is a logical NOT. The union of $B$ and $B'$ is an isomorphism from the hypernaturals to the relative integers under addition and multiplication.
(To convince yourself of this, let $A$ be the infinite sequence of 1s. If you apply the standard algorithm to add one to it, you carry the 1 infinitely and end up with an infinite sequence of 0s. So $A + 1 = 0$, and $A = -1$).
Anyway. This must have a name. What is it?