We study generating series of Gromov-Witten invariants of $E\times\mathbb{P}^1$ and their tropical counterparts. Using tropical degeneration and floor diagram techniques, we can express the generating series as sums of Feynman integrals, where each summand corresponds to a certain type of graph which we call a pearl chain. The individual summands are --- just as in the case of mirror symmetry of elliptic curves, where the generating series of Hurwitz numbers equals a sum of Feynman integrals --- complex analytic path integrals involving a product of propagators (equal to the Weierstrass-$\wp$-function plus an Eisenstein series). We also use pearl chains to study generating functions of counts of tropical curves in $E_{\mathbb{T}}\times\mathbb{P}^1_\mathbb{T}$ of so-called leaky degree.