Commutative monoid "generating" algebraic number field

Question

Let's say that we have an algebraic number field, which is also an ordered field (e.g. no roots of -1), and whose ring of integers is a UFD. We would like to form a multiplicative, commutative free monoid that "generates" this field in a certain sense. The monoid must have these properties:

1. The elements of this monoid are all $> 1$.
2. The monoid is generated by the set of all products of primes from the ring of algebraic integers built from our field, plus some basis that spans the group of units (except for $-1$).

Given such a monoid of "generalized natural numbers," we then generate the original algebraic field by

1. Building the set of quotients of these, forming a group under multiplication, and
2. Building the set of differences of these quotients, which is a field under addition and multiplication

(I'm not sure if the order matters here.)

If we do this starting with $\Bbb Q$, we get $\Bbb N$. If we do it starting with $\Bbb Q[\sqrt{5}]$, on the other hand, we get something nontrivial and interesting: a certain nontrivial subset of all numbers of the form $a + b\phi$ with $a, b \in \Bbb N$, made up of all products of primes in $\Bbb Z[\phi]$ and positive powers of $\phi$. And so on.

Questions:

1. When is this construction possible?
2. Does it satisfy any kind of interesting universal property? Something like: if this monoid embeds into any field, that embedding factors through this monoid's embedding into the original algebraic number field?
3. Similarly, when do we have any kind of stronger uniqueness property, in which we can uniquely derive this monoid from the original algebraic number field?

I think the answer to #3 is that we get uniqueness if and only if the group of units has rank = 1, otherwise we no longer have a unique basis for the group of units and thus a non-unique monoid in general.

This is building off of an answer to a related question. The original question was about semirings and was more subtle than I expected; some of the suggestions and comments in the answer seemed different and interesting enough to be a new question in its own right.