I am currently giving lectures in real analysis and a student asked an interesting question I couldn't answer, so I'm posting it here:

Let's say that a metric space $X$ is Baire if every countable intersection of dense open sets is dense. Then, by the Baire category theorem, every complete metric space is Baire. The converse doesn't hold literally, of course, because the Baire property is preserved under topological isomorphisms while the completeness is not. The question is whether it is the only obstacle, i.e.,

Is it true that every Baire metric space $(X,d)$ admits a metric $D$ generating the same topology such that $(X,D)$ is complete?