Correct form of object of sentence with grouped possessive and personal pronoun?

Lets say the object of a sentence is a possessive, of more than one nouns. Something that is say both someone else's and my own. IE Tom's and mine, as in the sentence

This meeting requires Tom and my's attendance.

How do I express this? Should I say "This meeting requires Tom and I's attendance" or "This meeting requires Tom's attendance and mine?" or maybe something else?

I've been taught "I" should only be used in the subject. However "Tom and I's attendance" seems most natural to me. I've used this form my whole life, but every time I type it I fear I may not be using it correctly.

I noticed this question, but it seems to apply to the subject, not the object.

Answer

This meeting requires Tom's and my attendance.

or

This meeting requires Tom's attendance and mine (or my own).