I agree with Ted's answer that there is an electric field present. However I think his answer is insofar incomplete, as it doesn't say how to measure the charge of the black hole. (I didn't understand the answer of Carl fully. How come the "restricted" electrons are still valid as GR test particles? Or if they are inside the BH, how does the information about the measurement come out?)
The thing is, (at least in first orders) the dynamics of an uncharged test particle orbiting a black hole stays the same, regardless if the black hole is charged or not. All that one would infer from observing the test particle is a different BH mass.
Aside, I don't know if we have observable charged test particles at hand, which would change the game. I would reason not, as if they should be observable, they either have to consist of gas, which would spread out due to mutual repulsion, or being a macroscopic object like a star or another BH, in conflict with being a test particle. A midsized/smaller star would probably be fine, but I don't think there is a charged one out there, as it would attract lots of opposite charge to be neutral.
In general, the test particle in the Schwarzschild spacetime has a Hamiltonian that can be transformed into an spherical harmonic oszillator, with decreased centrifugal force (see D.D'Orazio, P.Saha 2010, http://arxiv.org/abs/1003.5659). In Reissner-Nordström spacetime it is fairly similar: Increased mass decreases the centrifugal force, but interestingly, increased net charge decreases it (own calculations, not in the paper).
A hypothetical experiment could run as follows:
- Observe a test particle (e.g. galactic center star S2).
- Have a process at hand that drops (lots of) charge with know mass into the BH.
- The changed BH net charge will affect the orbit of the test particle as follows: If it increases the net charge of the BH, the (from test particle observation) inferred BH mass grows less than expected from the mass of the material dropped into the BH. If it decreases the net charge of the BH, we will have the opposite, the BH seems to be heavier than expected.
Let me know what you think.