It is impossible if the secondary is DELTA to get a Ground Reference from the primary to get 277 volts. The OP said the secondary had 277 volts to ground on all three phases
Wrong. It is not possible to get a
stable ground reference on a delta secondary, but you can produce a ground-referenced voltage of 277 on each line if you connect equal impedances from each line to ground. I suspect that is what has happened in the OP's case. (You can also create a more or less stable center point by connecting just the primary windings of a zig-zag transformer.)
You do not have a neutral point on the transformer side, but you do have a virtual neutral point created by a wye-connected load. If the center point is not connected to anything else, the delta will be ungrounded, but will have equal voltages from each line to the center point. If you ground that center point, then the line voltages will become ground referenced by that action even though no load current will be flowing through that ground.
Once the line-to-center loads become unequal for any reason, the center point will still be at ground, and still no current will flow through the ground. But the line to ground voltages will no longer be equal and some or all of the line to center loads will be damaged.
If you increase the load on just one line, the other two 277V loads will see increased voltages, potentially up to 480V.
If you decrease the load on one line, the other 277V loads will see reduced voltages and the load that you decreased will see a voltage above 277V, up to 480V.