Mild (consequences of) forcing axioms imply a general lifting theorem which allows to characterise precisely those homomorphisms between reduced products of countable structures which have a lift that factorises into homomorphisms between the factor structures. We will discuss recent extensions of such lifting theorems to the metric setting and ensuing rigidity phenomena for particular metric reduced product structures of interest. This is based on joint work with Ilijas Farah and Alessandro Vignati, with Alessandro Vignati, and with Andreas Thom.