  Teleportation is a facet where quantum measurements can act as a powerful
resource in quantum physics, as local measurements allow to steer quantum
information in a non-local way. While this has long been established for a
single Bell pair, the teleportation of a many-qubit entangled state using
non-maximally entangled resources presents a fundamentally different challenge.
Here we investigate a tangible protocol for teleporting a long-range entangled
surface code state using elementary Bell measurements and its stability in the
presence of coherent errors that weaken the Bell entanglement. We relate the
underlying threshold problem to the physics of anyon condensation under weak
measurements and map it to a variant of the Ashkin-Teller model of statistical
mechanics with Nishimori type disorder, which gives rise to a cascade of phase
transitions. Tuning the angle of the local Bell measurements, we find a
continuously varying threshold. Notably, the threshold moves to infinity for
the $X+Z$ angle along the self-dual line -- indicating that infinitesimally
weak entanglement is sufficient in teleporting a self-dual topological surface
code. Our teleportation protocol, which can be readily implemented in
dynamically configurable Rydberg atom arrays, thereby gives guidance for a
practical demonstration of the power of quantum measurements.
