We normally wrap names in <bdi> tags to ensure that they are properly displayed even if they contain RTL text, or a mixture of RTL and LTR. However, there are some places in HTML where <bdi> tags are not allowed; this concerns at least the page title (<title> tag) and tooltips (<a ... title="...">). See https://beta.musicbrainz.org/recording/01e5edba-5c8b-4290-a770-85a8ad820f48 for an example (the appearance is browser-dependent, but usually broken).

The recommended way to deal with these cases is to wrap the name between two special formatting characters, FSI (First Strong Isolate, U+2068) and PDI (Pop Directional Isolate, U+2069), which serve the same purpose as <bdi> for places where out-of-band markup is not available. They are a recent Unicode feature, and browser support is not so good yet (I have FF 31 and Chromium 37 available, neither supports it), so there is no hurry in implementing it; but we should do it at some point.