Oops, I missed the "anti".
I know for muon antineutrinos what I said still works, except maybe the year they were first detected. I don't remember whether those experiments specifically found neutrinos or antineutrinos. When I was a graduate student in the late 1970s and early 1980s I worked on an experiment at Fermilab that used a beam of muon antineutrinos. There was nothing special about producing them versus producing muon neutrinos. They're produced by letting negative versus positive pions or kaons decay.
According to the Web site of the experiment that first detected tau neutrinos, their apparatus produced both tau neutrinos and antineutrinos:
http://www-donut.fnal.gov/web_pages/DONUT/Design/Design.html
I don't see (yet) any specific reference to tau antineutrinos in their results. However, I don't see any reason why their detector would find only neutrinos and not antineutrinos. In order to distinguish them, you have to find out whether they produced a negative or postive tau lepton. I can't figure out yet whether they were able to distinguish between the two kinds of taus.