Are you certain that the sodium hydroxide wasn't the result of a chloralkali process? Electrolysis of sodium chloride brine is the primary production route for sodium hydroxide, and it also produces hydrogen and chlorine gas. However, this process involves the electrochemical oxidation of the chloride ions in solution, and the simultaneous reduction of water, thus producing hydroxide ions as hydrogen gas is liberated.
If it was obtained through a chloralkali process, then the process itself generated the necessary hydroxide ions from water, instead of them being present in the sea water originally. Perhaps I'm being too pedantic about the specifics -- I would similarly take issue if someone claimed that water was made of hydrogen and oxygen gas; it is absolutely made of hydrogen and oxygen atoms, but it takes a great expenditure of energy to convert them to their elemental forms.

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