Question Structure

[minamax]

Hello. A friend of mine has been learning some of the useful English phrases and expressions for use on the phone. When she came across the question "Who should I say is calling?", she got really confused. She had asked me what type of question it was. What was the subject of this question? Was this an embedded question? I really could not answer. So, I was wondering if anyone could explain this question and perhaps give a few other examples of this type of question.
Thank you in advance!

[Casiopea]

Welcome, minamax.

Underlying Structure: I should say who is calling
Subject-Verb Inversion: should I say who is calling?
WH-Movement: who should I say is calling?

Meaning
I should say who is calling.
I should tell e.g., my sister who is calling her.

Variation
Who may I ask is calling?
May I ask who is calling?

[minamax]

Dear Casiopea,

Thank you for your reply. I was wondering is any of the following would be alright. And if not, why not?

Who I should say is calling?
Who is calling should I say?
Who is calling I should say?

and what type of question is "Who should I say is calling?"

Thank you

[Casiopea]

[1] Who I should say is calling?
[2] Who is calling should I say?
[3] Who is calling I should say?

Examples [1] and [2] don't work because the subject-auxiliary set 'I should' looks like a statement. To get a question, you need to invert them, like this: I should => should I?

I'm not sure what you're asking here, 'what type of question is "Who should I say is calling?" ' The base structure looks like (A) below. (B) is the result of auxiliary+subject inversion, and (C) is the result of two things: (i) who movement, and (ii) auxiliary+subject inversion.

(A) I should say who is calling. <statement>
(B) Should I say who is calling? <question>
(C) Who should I say is calling. <question; requesting the person's name>

'who is calling' is the object of the verb 'say'. 'should' is a modal-auxiliary, and 'I' is the subject. The difference between (B) and (C) in terms of meaning is this. The answers:

(B) Yes, you should tell me the name of the person who is calling me.
(C) Yes. Please tell Max that Casiopea is calling him.

All the best.

[minamax]

Thank you so much, Casiopea!

After reading it a dozen of times... I finally got it! Thank you. Even before posting a question in this forum, I could understand the initial quesiton and what it meant, but I just could not figure out why it's written this way or how I could help my friend figure out the structure. Thank you for a clear breakdown!!!

[Casiopea]

You're welcome, minamax, and I'm sorry it took you a dozen times. (I'll have to work on that.)