First, although they say this is “very rare”, in the populations we often treat it is not.
Second, I spoke with an immunologist and asked how likely a false positive with an IgM is, and he said it’s such a large and unique molecule a true false positive IgM is exceedingly rare. (“Especially if symptomatic” he added.)
Third, Is the phenomenon studied anywhere? There are studies in Flaviviri (WNV) showing IgM can persist 6-8 years. [1-4] [I’ve had a fair amount of Flavi viri experience for a US doctor, and it is as weird as any of our worst chronically ill cases.] I don’t find much data in EBV research, but that doesn’t mean anything as ‘we’ see this in the chronically ill
Any idea why the persistence of IgM may be useful?
Yes. In mice anyway, it lowers mortality (having more IgM for a long time) which for protection against a nasty class like Flaviviridae can be useful, especially in endemic areas. 
Why does this happen?
Immunological: Autoimmunity underlying [even at low titer] can cause IgM persistence.
Cross-reactivity with other HHV [which chronically ill / fatigued folk keep around] will cause this as they rotate through ‘HHV of the month’… 
Toxicant exposure: I suspect Mycotoxins, chem and metal tox and even other unidentified infections can as well. [Studies in persistent IgM patients showed positive correlation to ‘substance use’ which is just toxicity under another name.]
Outside the viral world this also happens with bacteria / microbes that persist and can evade immune surveillance (cyst forms, other methods). Common to see it in C. pneumoniae, M. pneumoniae, Borrelia spp. and all its relatives. Biofilm disruption is also a common cause, in all types of infections. One of many MOA here would be
that neutrophils can trigger Ig differentiation and IgM  and biofilms have an intricate relationship with neutrophils.  (Side note, this is one of the most bad-ass biofilm-immunome papers I’ve read).
It also can happen after vaccination. 
Well, what do you really need? Along the ‘first do no harm’ line of investigation, lab error (although unlikely in IgM) should be considered. Next would be labs drawn too close together, although this is MUCH more an issue with things like EBV EaIgG or [d] which need 6-8 months between to be reliable.
For autoimmune work up at least run an ANA + Reflex. Also, any symptom specific labs including inflammatory markers and surrogates like ESR, CRP, D-dimer, Ferritin etc. Also, recall that low-titer ANA elevation almost always has a toxicant/chronic infection connection.
B-cell work-up: get total Ig G M A E and IgG 1-4 done. [I had one like this who had a selective IgA deficiency too.] I’d only do T-cell workup if you really suspect a CMI issue [I’ve seen maybe 5 (in three-decades) triggering something like this, so it can happen.]
Toxicants: I’d strongly recommend mycotoxin screening as well as chemical toxins + glyphosate and an unchallenged urine toxic metals normed to NHANES.
Cross reactive HHV: They are there so I just assume they are part of the problem. If you have unlimited money you can check all of them. Nowadays after running hundreds or more of HHV panels, I just tell people we are all full of HHV family – but sick folks let them be active and they take turns bothering your immunity, so testing can be a bit of “whack a mole”.
Other hidden infections, definitely if there is a history, or nothing else is making sense.
Biofilms: If sick over 60 days they have them. If sick over 1-year they have Phase-2 biofilms as well.
In this case, while there are the Flaviviridae data, not much else exists (I suspect because the larger ID community feels it’s unlikely to impossible). So, the above rely on the data that exists, and a great deal of clinical experience following such things. Also, I have had a reciprocal mentorship with an immunologist who actually invented early virology lab processes, and many of the modern HHV family lab platforms (it made him very wealthy). He’s older than me and used to hang out with Jonas Salk and that crew. I have access to his mind although he’s retired now, and he spent years learning nutrient – cell – immune
therapies from me, which I was amazed he found interest in.
Webinars to consider: 19, and then: 65, 23, 53