A central theme in quantum many-body physics is the emergence of universality classes. A universality class is often described by an effective quantum field theory (QFT), which can usually be characterized by a renormalizable UV Lagrangian. However, it has been realized that there exist "non-Lagrangian" QFTs which defy such a characterization. These non-Lagrangian theories can be viewed as ultra quantum matter that are not connected to any semi-classical mean-field limit, and it is fascinating to have them emerge in condensed matter systems. Motivated by this, I will discuss a general "emergibility problem": given a lattice system, by tuning its parameters can a specific QFT emerge in its ground state? I will argue that quantum anomaly provides powerful insight into this problem, based on which I will establish a systematic framework to study it. I will present the results obtained by applying this framework to the emergibility of Stiefel liquids. The Stiefel liquids form an infinite family of QFTs that unify the celebrated deconfined quantum critical point and Dirac spin liquid, and, more interestingly, generalize them to possibly non-Lagrangian theories. Our anomaly-based framework for the emergibility problem allows us to predict all different realizations and specific properties of the Stiefel liquids that can emerge in various frustrated magnets, such as triangular lattice spin-1/2 systems. Most of these realizations were not envisioned before, and some of them cannot be described using the conventional approach. These results provide useful guidance for creating non-Lagrangian theories in a lab.