Does anyone know of a recording or completion of Mendelssohn's last string quartet?
String Quartet in E, R 34-36 (1847)
Allegro vivace (1st or 4th movement), R36
The Scherzo and Andante sostenuto (R34-35) have been recorded as part of Op. 81 many times, of course.
Is there an actual completed quartet, similar to Schumann's violin sonata 3 vis a vis the well-known FAE sonata?
No - from what I understand there's a substantial movement (allegro vivace) that is unfinished. It received a number in the new complete Mendelssohn catalog (I sent a post to UC about this months ago). This would either be the first or (more likely) the fourth movement of a 4-movement work. Two other completed movements were published in the op. 81.