I ended up doing this in awk; it could probably be translated to perl by someone smarter than me. For some reason the filename is printed AFTER the results instead of before as I would have expected, but otherwise this seems to work.

BEGIN{FS="\|";}
{
   line= substr($0,0,length($0)-1); #peel trailing bar
   n=split($0,a);
   for (i=0;i<n;i++){
      line=line  a[i];
      if(a[i] ==""){nulls[i]++};
   }
}
END{
   print FILENAME;
   for (i in nulls) print "\t field " i": " nulls[i] " nulls";
}
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