Agreeing with the above.
It sounds like the receptacles have a good connection to the supply EGC and to earth ground, but that the transformer source is not properly connected to ground. Because the source is not ground referenced, any test between the receptacle powered terminals and receptacle ground pin would show 'floating ground', even though the real problem is 'floating source'.
A good solution would be to replace the transformer with one that could have a grounded terminal. If your equipment only requires straight 240V, then either a secondary 240V delta with grounded center tap, or a secondary 240/139V wye would be fine.