ca.analysis and odes - Is a function which is finitely multiple-valued in each variable separately, also finitely multiple-valued in all its variables jointly?

It is well known that under suitable conditions, a function which is:

1. a polynomial in each variable separately is a polynomial in all its variables jointly.




2. a rational function in each variable separately is a rational function.




3. a holomorphic function in each variable separately is holomorphic in all its variables.

A complete analytic function can be single-valued or multiple-valued according as it does not have, or does have, branch points. The algebraic functions are examples of the latter.

Here is my question: is a complete analytic function, which is finitely multiple-valued in each variable separately, also finitely multiple-valued jointly?