What is the hausdorff completion of this uniform structure on the real line?

Question

Take the real line $\mathbb R$. Instead of the usual uniform structure on $\mathbb R$, we will say that each neighborhood of the diagonal of $\mathbb R \times \mathbb R$ is an entourage. This defines a uniform space.

This space definitely is not complete. What is this space's Hausdorff completion? (A follow up question is to ask the same thing for $\mathbb R^n$.)

Answer 1

As $\mathbb{R}$ is paracompact, the collection of neighbourhoods of the diagonal of $\mathbb{R} \times \mathbb{R}$ indeed is a uniformity.

In this question and its nice answer by Brian M Scott it is shown that this uniformity is already complete.