A graph G is a homomorphic preimage of another graph H, or equivalently G is H-colorable, if there exists a graph homomorphism f : G → H. A geometric graph Ḡ is a simple graph G together with a straight line drawing of G in the plane with the vertices in general position. A geometric homomorphism (respectively, isomorphism) Ḡ → H̄ is a graph homomorphism (respectively, isomorphism) that preserves edge crossings (respectively, and non-crossings). The homomorphism poset ℊ of a graph G is the set of isomorphism classes of geometric realizations of G partially ordered by the existence of injective geometric homomorphisms. A geometric graph Ḡ is ℋ-colorable if Ḡ → H̄ for some H̄ ∈ ℋ. In this paper, we provide necessary and sufficient conditions for Ḡ to be 𝒫n-colorable for n ≥ 2. Along the way, we also provide necessary and sufficient conditions for Ḡ to be 𝒦2,3-colorable.