We define an abstract complex saddle loop in $\mathbb C^2$ as a pair $(F,R)$ of a hyperbolic normalized saddle foliation $F$ with a corner Dulac map D and a regular map $R\in\mathrm{Diff}(\mathbb C,0)$. Up to an appropriate equivalence relation that corresponds to different determinations of complex Dulac and to transversal changes, the first return map is given by $F=RD$ on the universal cover of the standard quadratic domain. We show that such Poincar` e maps are rigid, in the sense that their non-ramified formal conjugacy implies the analytic conjugacy.