Active feedback between geometry and physics is woven throughout the study of Nature at its fundamental level, and is of key importance in string theory. Methods of complex algebraic geometry in particular have brought about an unrivaled abundance of solutions, counted well into hundreds of orders of magnitude, reciprocated by the discovery of the wholly unexpected mirror symmetry. However, recent developments demonstrate that there are rich additional possibilities, made possible by certain generalizations that, at first glance, appear to be non-algebraic. Nevertheless, they are remarkably well-aligned within an overall mirror-symmetric framework, are amenable to almost as comprehensive quantitative analysis, and hint at a deeper relationship with symplectic geometry.