Factorization of rings of integer-valued rational functions

$\DeclareMathOperator{\Int}{Int}\DeclareMathOperator{\IntR}{Int{}^\text{R}}$For a domain $D$, the ring $\Int(D)$ of integer-valued polynomials over $D$ is atomic if $D$ satisfies the ascending chain condition on principal ideals. However, even for a discrete valuation domain $V$, the ring $\IntR(V)$ of integer-valued rational functions over $V$ is antimatter. We introduce a family of atomic rings of integer-valued rational functions and study various factorization properties on these rings.