L1-N = 120v 1ph
L2-N = 120v 1ph
L1-L2 = 208v 1ph
There is no equipment that will care about the phase angle, which only manifests itself as a voltage difference.
One corner case that has been seen to cause problems in the real world is in some cooking equipmen that uses electrical heat from a sealed element, such as coffee makers. The design of the product often allows for a certain amount of capacitive (and maybe even resistive?) current leakage from each portion along the length of the heating element to grounded metal.
When the unit is driven from 120-0-120, even with no neutral connection to the element, the capacitive leakage current from L1 is roughly equal to and opposite the leakage current from L2. The result is that the apparent ground leakage current is reduced by an order of magnitude or so and the unit does not trip a GFCI, either receptacle or breaker type.
But when L1 and L2 are two-out-of-three from a three phase source, producing 208 volts across the element, the reduction in heating is tolerable. But the visible ground leakage current is now the vector sum of two components which are at a 120 degree phase angle instead of a 180 degree phase angle. And the GFCI will invariably trip.
If the metal shell of the appliance is not grounded, the GFCI may not trip, but a person touching the metal and a ground will get a noticeable shock.
Similar problems in residential use are less common because most kltchen applicances are powered from 120V and so their leakage in use will correspond to the leakage during testing.
