  We consider determinantal ideals, where the generating minors are encoded in
a hypergraph. We study when the generating minors form a Gr\"obner basis. In
this case, the ideal is radical, and we can describe algebraic and numerical
invariants of these ideals in terms of combinatorial data of their hypergraphs,
such as the clique decomposition. In particular, we can construct a minimal
free resolution as a tensor product of the minimal free resolution of their
cliques. For several classes of hypergraphs we find a combinatorial description
of the minimal primes in terms of a prime splitting. That is, we write the
determinantal ideal as a sum of smaller determinantal ideals such that each
minimal prime is a sum of minimal primes of the summands.
