Apologies for the long delay in getting this analysis done. My full write up is on MediaWiki.

Summary:

• The different phonetic algorithms vary widely in terms of how aggressive they are about grouping more-or-less similar words together, in part because they have different goals, and were created at different times (the oldest going back ~100 years!).
• They generally have difficulty with words with numbers, short words, and words with diacritics. Most of these problems can be ameliorated with additional filters in the analysis chain.
• Double Metaphone seems like the best of the bunch for the next phase of testing.

Next Steps (see T184771):

• Implement and test an analysis chain with the obvious precision mitigation steps: character folding, stop word filtering, filtering words with numbers, and word length filtering.
• Set up a RelForge test bed for English Wikipedia and Wiktionary with a keyword or other implementation for phonetic title/redirect search or other implementation method, and invite feedback via the Village Pump, mailing lists, etc.
  • Depending on feedback, possibly adjust encoding, code length, etc., and test again.
• If all goes well, make a deployment plan—using a keyword, or making it a component of standard search (which then has to interact with Learn-to-Rank), etc.