Is there advantage to balanced phono-preamp

If you don't have a fully balanced system you are not gaining anything by adding balanced phono corrector.

Is there a "balanced" signal coming from the cartridge?

There seems to be some confusion here.

A phono cartridge is inherently balanced. That is, it has separate hot and neutrals for left and right, and one shield.

There are no better places to run balanced connections than from your phono cartridge into your phono preamp, and then from your phono preamp to your line stage. If those are truly balanced (differential) amplifiers, then balanced connections should yield the highest gain and the lowest noise.

If the "balanced" amplifiers aren't really differential amplifiers - but use op-amps on the inputs and outputs to process balanced connections - then you may lose some of the benefits of a truly balanced set of connections. But with phono connections, it's still likely to result in the lowest noise.

Do you know of a manufacturer who makes a balanced XLR-XLR phono cable, or is any balanced cable going to work well?

Whether it's a moving coil type or a moving magnet type or any of the variants thereof, the signal comes off a coil of wire.  At either end of the coil, the signal looks the same, except for its being opposite in phase.  So, a cartridge is an inherently balanced output device.  When you run it into a typical standard single-ended phono stage, you make an arbitrary choice to take the signal off one end of the coil and to ground the other end.  But there is no need to connect to ground.  In balanced mode, you just connect the coil to another identical circuit.  Because the two signals are 180 degrees out of phase, any noise common to both signals is cancelled.  You get a bit more gain in balanced mode and that advantage of noise cancellation, known as Common Mode Rejection.
Brrgrr, usually you won't want an XLR to XLR phono cable. Usually you will want a DIN to XLR cable, to connect your cartridge to a balanced phono input.  Yes, they are made, can often be custom-made too.