I made a simulation. By grounding one end of the AC source, you eliminate outputs 3 and 4.
I removed the ground at the AC source. Now all 4 outputs show a signal. These are out of phase, each 90 degrees from the next. I believe this is what you want.
You still need the two zero grounds at the RC network. I removed them, and suddenly all four outputs became nearly in phase with each other.
The circuit as O/P posted works correctly, after allowing for the start-up transients for the simulation. If the ground on the 1 tone source is removed then signals at the output are still present but at different amplitudes. Two of the output signals are missing if the 1 tone source +ve pin is connected to the common-mode pin (c) of the balun. The +ve pin should be connected to the differential mode pin (d).
Looking at the VspecTran measurement may be confusing though as it only shows the magnitude of the signal and not the phase.
Actually for this application with a grounded common-mode pin to the balun it is recommended in the documentation to use the Balun3Port instead of the Balun4Port as it provides "an equivalent, but numerically more robust" result. Both work in this example though.