  We prove that the moduli stack of Bridgeland semistable holomorphic triples
over a curve of $g(C)\geq 1$ with a fixed numerical class and phase is an
algebraic stack of finite type over $\mathbb{C}$ and admits a proper good
moduli space. We prove that this also holds for a class of Bridgeland stability
conditions on the category of holomorphic chains $\mathcal{T}_{C,n}$.
  In the process, we construct an explicit geometric realisation of
$\mathcal{T}_{C,n}$ and prove the open heart property for noetherian hearts in
admissible categories of $D^b(X)$, where $X$ is a smooth projective variety
over $\mathbb{C}$, whose orthogonal complements are geometric triangulated
categories.
