How does the cathode of an electrolytic cell drive the anode reaction?

Consider this electrolytic cell:


![Electrolytic Cell](https://i.stack.imgur.com/pqVPF.png)


In this cell, electrons from the anode of the battery travel to the cathode of the electrolytic cell, where hydrogen ions are reduced into hydrogen gas. This drives the oxidation reaction at the anode, where hydroxide ions are oxidized into oxygen gas.


My question is: How does the reduction reaction at the cathode drive the oxidation reaction at the anode? They aren't connected by a wire or anything, so there is no electron flow from the cathode to the anode like in a galvanic cell. Also, since this is an electrolytic cell, the reaction is non-spontaneous.


Some possible hypotheses:


1. The galvanic cell just gave some of its electrons away, so it is negatively charged and sucks in the electrons from the anode of the electrolytic cell, forcing the oxidation reaction to occur. However, assuming the galvanic cell has a salt bridge, shouldn't ions be able to compensate for this?
2. The loss of positively charged ions at the cathode causes a buildup of negative charge in the cell; the anode tries to compensate for this. However, how do electrolytic cells involving the reduction of water, which has neither a positive or negative charge, into hydrogen gas occur?


