Supplimentary Information

The Supplementary Information includes details regarding the UV-regularized models discussed in this work. Details on the implicit Euler-Maruyama scheme used to integrate the auxiliary fields in simple fluids is provided. Detailed derivations and notes on integrating the density fields using the variable transformed $\Phi$-Euler-Milstein scheme are provided. Additional numerical and analytical analysis for the monatomic ideal fluid are provided that give some guidance regarding the failure discussed in the main text.