Let $A=\{a_0, a_1, \ldots\}\subseteq\mathbb{R}$ and $\Gamma$ be the set of functions $f$ from the natural numbers $\omega$ (including $0$) into $A$ such that $f(n)\in\{a_{2n}, a_{2n+1}\}$. Now let $P$ be the set of functions from $\Gamma\times\omega^{<\omega}$ into $\{0, 1\}$ (where $\omega^{<\omega}$ is the set of finite sequences of $\omega$). For natural numbers $n_1, \ldots, n_N$ and $f\in\Gamma$, write $(-1)^\pi f(n_1)\cdots f(n_N)$ in place of $(-1)^{\pi(f, n_1, \ldots, n_N)}f(n_1)\cdots f(n_N)$ for brevity (the product of $f(n_1), \ldots, f(n_N)$ up to sign).
I am looking for an $A$ with $a_0<a_1<\cdots$ such that for each natural numbers $n_1<\cdots<n_N$ and $m_1<\cdots<m_M$, for each finite subset $G$ of $\Gamma$, and for each $\pi\in P$, I have $\sum_{f\in G}(-1)^\pi f(n_1)\cdots f(n_N)=\sum_{f\in G}(-1)^\pi g(m_1)\cdots g(m_M)$ if and only if $N=M$ and $n_1=m_1, \ldots, n_N=m_N$.
If $G$ is limited to singletons only, then $A$ can be taken to be the set of primes. Apart from that, I am still wondering (if that even make sense). I'm interested along the lines of generalizing the trick of unique prime factorization to also work for addition.
I think the property of the sequence of primes you want to generalise can be more simply expressed using a bit more abstract algebra like this:
I think you are asking if you can replace the sequence of primes by some other sequence of numbers and get a mapping that is one-to-one for sums of signed monomials. The answer is yes: the field of real numbers has infinite transcendence degree over the rationals. This implies that you can find a sequence of real numbers $a_0, a_1, \ldots$, such that the mapping $X_i \mapsto a_i$, $i = 0, 1, \ldots$, induces a one-to-one homomorphism $h$ from $\Bbb{Q}[X_0, X_1, \ldots]$ to $\Bbb{R}$, i.e., a mapping that is one-to-one on arbitrary sums of monomials.