Outcome depends upon the details of attachment. Imagine vacuum tube diode with two tiny and almost transparent voltmeter electrodes immersed in electron flow between cathode and anode. Let us set one of voltmeter electrodes even much more transparent than another. Then electrons will be accumulated preferrably on less transparent making it negatively charged.

A generic nonequilibrium context could be given by a jar with some solvent and a barrier in the middle. On the right you have an high concentration of a solute, and on the left just the solvent. The heigth of the liquid is the same for the two regions, in a similar fashion to the flat chemical potential in the generic nonequilibrium system you were describing, but if you pull out the barrier you obtain transport of the solute from right to left, destroying the order. An ideal intrument could measure the difference of concentration, so I suppose that you could do the same in your system and measure a voltage difference. The only problem, as underlined by Dr. Fedichkin, would be the effects of the same electrodes on your sytem.

In general, we understand that the voltmeter measures a "chemical potential" of charged species of particles. Of course, two systems are said to have equal chemical potentials when they no longer exchange particles with each other. Indeed, this is how a chemical potential is defined in statistical mechanics. Furthermore, an ideal measurement doesn't disturb the state of the system (this is usually encoded in "ideal voltmeters have infinite impedence).

However, the trouble with a general quantum system (open quantum systems) is that they exchange both energy and particles. This means, one cannot measure temperature [ Remember, temperature is defined to be the equal for two systems which no longer exchange energy with one another] and voltage separately for such systems. 

One can actually prove theorems which effectively say "that measuring only voltage (by requiring that the probe exchange no particles system) or measuring only temperature (by requiring that the probe exchange no energy with the system) is quite meaningless in characterizing the thermodynamic state of the system you are measuring." One needs to measure both these quantities together.

This has been shown for fermions and for nonequilibrium steady states which can be very far from equilibrium [Of course, results hold at equilibrium as well because it is a special case of a nonequilibrium system which is really close to equilibrium]. A very important thing to remember is that the probe must be noninvasive (weakly coupled to system of interest). 

Showing these results for Bosons is nontrivial. Recent work has shown this for Fermions, for nonequilibrium steady states and noninvasive probes.

[ https://arxiv.org/abs/1603.00096]