To explore the concept further, note that the overall charge density of the wire is in fact something that depends on your frame of reference.
To see this, imagine that you're on a starship next to an infinitely long, neutrally-charged but current-carrying wire. As you accelerate closer to the speed of light, you "catch up" to the electrons moving in the wire -- but conversely, the 'stationary' positive ions they're loosely attached to are now moving backwards.
Since they're moving relative to you, they now experience length contraction; meanwhile, the electrons, which were already moving and therefore already contracted relative to your original inertia frame, are spreading back out as you catch up. So the relative charge densities will themselves change as you move -- meaning that the wire appears to pick up a charge!
And, of course, whether you're moving relative to the wire or it's moving relative to you doesn't matter -- so you immediately see that yes, a moving neutrally-charged wire should general produce an electric field.
(This is somewhat separate to the Ampere's Law effect from Dale, though they turn out to be the same law when you move into the full spacetime formalism.)