In the recursive lamination of the disk, one tries to add chords one after an other at random; a chord is kept and inserted if it does not intersect any of the previously inserted ones. Curien and Le Gall [Ann. Probab., vol. 39, pp. 2224--2270, 2011] have proved that the set of chords converges to a limit triangulation of the disk encoded by a continuous process $\mathscr M$ . Based on a new approach resembling ideas from the so-called contraction method in function spaces, we prove that, when properly rescaled, the planar dual of the discrete lamination converges almost surely in the Gromov--Hausdorff sense to a limit real tree $\mathscr T$, which is encoded by $\mathscr M$. This is one of the first natural limit real trees which is identified and does not come from the excursion of a L{é}vy process.