This talk will be concerned with some aspects of the renormalization of the $\Phi^4$ stochastic partial differential equation, in the singular but subcritical (also called super-renormalizable) range. Recently, a novel perspective on the theory of regularity structures, based on multi-indices rather than trees, has been introduced. This will be the viewpoint taken in this talk. I wish to first introduce the notion of "model", which in this context may be motivated by considering the "geometry" of the solution manifold. Because this model is indexed by multi-indices, which form a much sparser index set than trees, it turns out that Hairer's original fixed-point formulation is not applicable in order to construct an actual solution to the equation from this model. In that regard, I would like to describe a more "intrinsic" strategy for well-posedness which does not rely on a fixed-point formulation and which we implement in a space-time periodic setting. (Based on joint works with Felix Otto, Rhys Steele and Markus Tempelmayr)