The multiphysics modelling of tungsten inert gas (TIG) welding requires the numerical interaction between different physics and multiple conjugate bodies. In simplified and single domain TIG models, the behaviour of the partitioned multiphysics algorithms is generally stable and satisfactory. However, the increase of more complex and multi-domain TIG welding models introduces issues with algorithm convergence. In fact, because the coupling of conjugate domains, modelled with different physical variables, is also generally performed in a partitioned manner, the coupling algorithms can be long and/or difficult to converge. Therefore, increasing the physical complexity of multiphysics TIG welding models necessitates the careful comparison of partitioned and fully/quasi-monolithic algorithms. Thus, this motivates our investigation of mixed-variable fully/quasi-monolithic methods that couple conjugate bodies, modelled with different physical variables, in a more robust manner. To this end, we investigate the implementation of asymmetric constraints to couple non-linear and multivariate interactions between conjugate bodies. In particular, we encounter the need for asymmetric constraints when looking to strongly couple the thermo-hydraulic interactions between the arc-plasma and that of the melting workpiece. This manifests in the conjugate heat transfer and phase change problem because the temperature variable is used to model the arc-plasma while the enthalpy variable is used to model the melting workpiece. Numerically, the asymmetric constraints allow for a monolithic and quasi-strong coupling of the temperature based arc model to the enthalpy based workpiece model, which results in a conservative algorithm that weakly imposes thermal continuity. To benchmark the performances of the mixed variable monolithic approach, both partitioned and fully/quasi-monolithic algorithms are implemented for the conjugate heat transfer and phase change problem. The performances of the fully/quasi-monolithic algorithms, constructed using the asymmetric multipliers, are compared to those of the analogous but fully partitioned Dirichlet-Neumann thermal algorithm. The performances of the algorithms constructed using the asymmetric constraint method make the novel algebraic implementation of interest to the TIG welding and multiphysics modelling community.