Which spectra arise from partially ordered commutative monoids? Thomason showed how any connective spectrum arises from a symmetric monoidal category:

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*Robert W. Thomason, Symmetric monoidal categories model all connective spectra, Theory Appl. Categ. 1 (1995), 78–118.

A special case of a symmetric monoidal category is one whose underlying category is a poset.   Such a thing is both a commutative monoid and a poset, with the property that
$$ x \le x'  \textrm{ and } y \le y' \; \implies \; xy \le x'y' $$
Let me call these things 'partially ordered commutative monoids'.
If we restrict Thomason's construction to partially ordered commutative monoids, which spectra do we get?
The first step of Thomason's construction is taking the geometric realization of the nerve of a symmetric monoidal category and getting an $E_\infty$ space.  The second is group completion.
On the one hand, every homotopy type is weakly equivalent to the geometric realization of the nerve of some poset.  Indeed something stronger is true: every simplicial complex is homeomorphic to the geometric realization of the nerve of some poset.
But on the other hand, the $E_\infty$ space we get by taking the geometric realization of the nerve of a partially ordered commmutative space is always a commutative topological monoid – commutative 'on the nose', not just up to coherent homotopy.
I don't know if we get every commutative topological monoid (up to some relevant notion of equivalence) as the geometric realization of the nerve of a partially ordered commutative monoid.
I will however venture a guess, just to prompt someone to prove, disprove or improve it.  I'll guess that we can get any connective Eilenberg-Mac Lane spectrum by applying Thomason's procedure to a partially ordered commutative monoid.
 A: Contrary to my claims in the comments and in my other answer, the only connective spectra which arise as the group completion of a partially ordered commutative monoid are the discrete ones -- $HA$ for a discrete abelian group $A$. Here is a proof.
Observation: Let $A$ be a partially-ordered abelian group. Then the canonical map $U : A \to \pi_0(A)$ is a weak homotopy equivalence.
Proof: It will suffice to show that $U$ has weakly contractible fibers. Since the fibers are all isomorphic, it suffices to look at the fiber over $[0] \in \pi_0(A)$. That is, it will suffice to treat the case where $A$ is connected (and show that $A$ is weakly contractible).
Note that if $x \geq 0 \leq y$, then $x \leq x + y \geq y$. So more generally, if $x \geq z \leq y$, then $x \leq x + y - z \geq y$. This observation allows us to reduce any zigzag in $A$ to be of length 1. Since $A$ is connected, this means that for any $x \in A$, there exists $y \in A$ such that $0 \leq y \geq x$.
Now consider the inclusion $i : A_{0 \leq} \to A$. Since $A_{0 \leq}$ has an initial object, it is contractible, so it will suffice to show that $i$ is a weak homotopy equivalence. By Quillen's theorem A, we are reduced to checking that for every $a \in A$, the category $a \downarrow i = A_{a \leq} \cap A_{0 \leq}$ is weakly contractible. This is equivalently the category of spans $(0 \leq x \geq a)$ in $A$. By the previous paragraph, this category is nonempty. Moreover, it has "functorial amalgamation" -- if $(0 \leq x \geq a)$ and $0 \leq y \geq a$, then both elements are $\leq$ to $(0 \leq x + y - a \geq a)$. By the Lemma below, this implies that $A_{0 \leq} \cap A_{a \leq}$ is weakly contractible as desired.
Theorem: Let $P$ be a partially-ordered commutative monoid. Then the group completion of $P$ is $HA$, where $A = K_0(\pi_0(P))$ is the Grothendieck group of its connected components.
Proof: Let $P$ be a partially-ordered commutative monoid. Then for any $p \in P$, the braiding map $p \otimes p \to p \otimes p$ is homotopic (in fact, equal) to the identity. By a certain group-completion theorem [1], this is a sufficient condition to guarantee that $P[p^{-1}] = colim(P \xrightarrow p P \xrightarrow p P \xrightarrow p \cdots)$ [2]. That is, the universal $E_\infty$-space receiving an $E_\infty$ map from $P$ which carries $p$ to an invertible element can be computed via the indicated filtered colimit. It follows that for any $p_1,\dots,p_n \in P$, we can compute $P[p_1^{-1},\dots,p_n^{-1}] = P[(p_1 \otimes \cdots \otimes p_n)^{-1}]$ by such a filtered colimit.
Consider now the case where $P$ is is finitely-generated as a commutative monoid, with generating set $p_1,\dots,p_n$. Then the group completion $P[P^{-1}]$ of $P$ is computed via the above formula, as $P[P^{-1}] = colim(P \xrightarrow{p_1\otimes \cdots \otimes p_n} P \xrightarrow{p_1\otimes \cdots \otimes p_n} \cdots)$, and is still a partially-ordered commutative monoid. So $P[P^{-1}]$ is a partially-ordered abelian group. By the Observation $P[P^{-1}] = H(K_0(\pi_0(P)))$ as desired.
Because group completion commutes with filtered colimits, and because $P$ is the filtered colimit of its finitely-generated partially-ordered submonoids [3], the result now follows for general $P$.
Lemma: Let $C$ be a nonempty category with functorial amalgamation in the sense that there is a functor $T : C \times C \to C$ and natural transformations $\pi_0 \Rightarrow T \Leftarrow \pi_1$, where $\pi_0,\pi_1 : C \times C \rightrightarrows C$ are the projection functors. Then $C$ is weakly contractible.
Proof: Pick $c \in C$. Then $T(c,-)$ admits a natural transfomration from the constant map at $c$, and also a natural transformation from the identity map on $C$. Upon geometric realization, then the constant map is homotopic to $T(c,-)$, which is homotopic to the identity, so $C$ is weakly contractible as desired.
[1] As discussed e.g. here -- a more general sufficient condition is that the cyclic permutation $p \otimes p \otimes p \to p \otimes p \otimes p$ be homotopic to the identity. This condition was isolated in unpublished work of Jeff Smith in his study of Adams' work on the smash product of spectra, and popularized by Voevodsky in his work on motivic homotopy theory.
[2] Weak equivalences of simplicial sets are stable under filtered colimits, so this filtered homotopy colimit may be computed as a colimit of simplicial sets without any cofibrant replacement. Moreover, posets are closed under filtered colimits of simplicial sets, so in fact this colimit may be regarded as a filtered colimit of posets.
[3] See [2].
