In this talk, we focus on phylogenetic inference using algebraic and semi-algebraic invariants of single-site substitution models used in Markov processes over phylogenetic trees. These models give rise to algebraic varieties, the study of which has proven effective for phylogenetic reconstruction. However, not all points in these varieties correspond to biologically meaningful or probabilistically valid scenarios, so it seems necessary to consider the subsets that do. In this talk, we explore the structure and meaning of these semi-algebraic subsets and demonstrate their important role in phylogenetic inference. We also present two reconstruction methods, Semi-Algebraic Quartet Reconstruction (SAQ) and Algebraic and Semi-Algebraic Quartet Reconstruction (ASAQ), which are consistent with the most general Markov model of nucleotide substitution and allow for rate heterogeneity across lineages. Despite their theoretical foundation, the use of these algebraic and semialgebraic constraints in phylogenetic inference poses significant challenges in practice. We will discuss these challenges and offer ideas for bridging the gap between theoretical results and practical applicability.