I can answer your question with the specialization to *convex* polyhedra and polytopes.
Specializing further to $\mathbb{R}^3$, the result is that

$n \ge 4$ positive real numbers are the face areas of a convex polyhedron
if and only if the largest number is not more than the sum of the others.

I wrote up a short note
establishing this: "Convex Polyhedra Realizing Given Face Areas," arXiv:1101.0823.
The result relies on Minkowski's 1911 theorem, which perhaps you know:

**Theorem (Minkowski)**. Let $A_i$ be positive faces areas and $n_i$ distinct,
noncoplanar unit face normals,
$i=1,\ldots,n$.
Then if $\sum_i A_i n_i = 0$, there is a closed convex polyhedron
whose faces areas uniquely realize those areas and normals.

This theorem reduces the problem to finding orientations $n_i$ so that vectors of
length $A_i$ at those orientations sum to zero. And this is not difficult.
Here is Figure 3 from my note from which you can almost infer the construction:

Minkowski's theorem generalizes to $\mathbb{R}^d$ and so does an analog of the above claim
(but I did not work that out in detail in the arXiv note).
In terms of an algorithm, the decision question is linear in the number $n$ of facet areas,
and even constructing the polyhedron is linear in $\mathbb{R}^3$,
and likely $O(dn)$ in $\mathbb{R}^d$ (but again, I didn't work that out).

But you don't mention the word "convex" in your post, so perhaps you are interested
in nonconvex polyhedra and polytopal complexes?