Suppose I have a complete set of two commuting observables $\hat{A}, \hat{B}$ for which:
$$ \hat{A} |{a} \rangle = a |a\rangle $$
$$ \hat{B} |b \rangle = b |b\rangle $$
Now, I can find a common eigenstate in the Hilbert space where $\hat{A}$ and $\hat{B}$ operates, or I can define a new Hilbert space which is the direct product of the space spanned by $|{a} \rangle$ and $|{b} \rangle$:
$$\{ |a,b \rangle \}_{H} = \{ |a \rangle \}_{H_1} \otimes \{ |b \rangle \}_{H_2}$$
Where $H$ are Hilbert spaces, and with the operators extension $A \to A\ \otimes \ 1$ ; $B \to 1\ \otimes \ B$.
In the new Hilbert space I can then choose the common eigenbasis as $|{a} \rangle \otimes|{b} \rangle$.
Now there is a rationale in Quantum Mechanics to choose one method or another, or is a matter choice ? And if indeed is a matter of choice, what are the situation in which one may prefer a method over another?