Connecting center-tapped transformer to earth ground; or, why am I being electrocuted?
That's a big fat unscreened transformer, so there is plenty of capacitive coupling between the input and output. A high impedance meter could easily read a fraction of mains voltage between the center tap and earth. If you flip the input connections you'll likely get a somewhat different measurement. 

The measure of health of the transformer is the leakage current at near maximum isolation voltage. It's not possible to measure that easily with a simple multimeter. The voltage measurement alone is not an indication of a problem, however the tingle could be. It would only take microamps to make the meter read something like 77VAC, but a perceptible tingle would reportedly take more like 1mA. 

There is no problem in earthing the secondary center tap- if it blows a fuse, the transformer is faulty.