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Building upon recent results of Dub\'edat (see arXiv:1403.6076) on theconvergence of topological correlators in the double-dimer model considered onTemperleyan approximations $\Omega^\delta$ to a simply connected domain$\Omega\subset\mathbb C$ we prove the convergence of probabilities ofcylindrical events for the \emph{double-dimer loop ensembles} on$\Omega^\delta$ as $\delta\to 0$. More precisely, let$\lambda_1,\dots,\lambda_n\in\Omega$ and $L$ be a macroscopic lamination on$\Omega\setminus\{\lambda_1,\dots,\lambda_n\}$, i.e., a collection of disjointsimple loops surrounding at least two punctures considered up to homotopies. Weshow that the probabilities $P_L^\delta$ that one obtains $L$ after withdrawingall loops surrounding no more than one puncture from a double-dimer loopensemble on $\Omega^\delta$ converge to a conformally invariant limit $P_L$ as$\delta \to 0$, for each $L$.  Though our primary motivation comes from 2D statistical mechanics andprobability, the proofs are of a purely analytic nature. The key techniques arethe analysis of entire functions on the representation variety$\mathrm{Hom}(\pi_1(\Omega\setminus\{\lambda_1,\dots,\lambda_n\})\to\mathrm{SL}_2(\mathbbC))$ and on its (non-smooth) subvariety of locally unipotent representations.In particular, we do \emph{not} use any RSW-type arguments for double-dimers.  The limits $P_L$ of the probabilities $P_L^\delta$ are defined ascoefficients of the isomonodormic tau-function studied by Dub\'edat withrespect to the Fock--Goncharov lamination basis on the representation variety.The fact that $P_L$ coincides with the probability to obtain $L$ from a sampleof the nested CLE(4) in $\Omega$ requires a small additional input, namely amild crossing estimate for this nested conformal loop ensemble.

Tau-functions \`a la Dub\'edat and probabilities of cylindrical events for double-dimers and CLE(4)