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In this paper, we introduce relative left Bongartz completions for a givenbasic $\tau$-rigid pair $(U,Q)$ in the module category of a finite dimensionalalgebra $A$. They give a family of basic $\tau$-tilting pairs containing$(U,Q)$ as a direct summand. We prove that relative left Bongartz completionshave nice compatibility with mutations. Using this compatibility we are able tostudy the existence of maximal green sequences under $\tau$-tilting reduction.We also explain our construction and some of the results in the setting ofsilting theory.

Relative left Bongartz completions and their compatibility with mutations