Let $P$ be a pseudometric space, and let $M_P$ be the metric space obtained by quotienting out points separated by zero distance. We can always complete $M_P$ to $\overline{M_P}$ by forming the pseudometric space of Cauchy sequences $\operatorname{pre}(\overline{M_P})$ and then quotienting out sequences that differ by a null sequence. The completion $\overline{M_P}$ is canonical in the sense that all other complete metric spaces into which $M_P$ embeds densely are isometric to $\overline{M_P}$ in a way the respects the embeddings. $\overline{M_P}$ also satisfies a universal property with respect to $M_P$.
Alternatively, we can construct $\overline{M_P}$ by first forming the complete pseudometric space $\overline P$ of Cauchy sequences in $P$, and then quotienting in appropriate ways to reach $\operatorname{pre}(\overline{M_P})$ and $\overline{M_P}$.
Do $\overline P$ and $\operatorname{pre}(\overline{M_P})$ bear any canonical relationship to $P$ in a similar way that $\overline{M_P}$ does to $M_P$ ?
Motivation: The $\mathcal L^p$ spaces are complete pseudometric spaces, and I think they are the completion of the pseudometric space of step maps, so it would be helpful to understand if this relationship is canonical.