Syncing mail on iPhone15 and MacBook Air M3

In Mail, you can use File > New Mailbox to create a new mailbox. Use the pop-up menu in the dialog to create it under the section called On My Mac.

You can then select messages and "Move to" or "Copy to" the new mailbox. Once you select a message, there are several ways to choose these options, including

• Using Message > Move to or Message > Copy to
• Right-clicking on the message summary in the middle column to get a contextual menu
• Clicking on "Move to…" or "Copy to…" in the Toolbar. ("Copy to…" is hidden by default, but you can customize the Toolbar using View > Customize Toolbar…)

If both are connected to the same e-mail account, in the usual way, that's normal. The master copy of the e-mail lives on your e-mail provider's servers. Synchronization means synchronizing the deletes, not just the fetching of new mail.

You could use the Mail application on your Mac to copy mail from your regular mailbox to an "On This Mac". (Then, of course, it will be your responsibility to back up your Mac.)

Yes, it does mean that you must move each message that you want to keep hidden from your iPhone. I assume that you know that you can select multiple messages at one time and issue a Move command that applies to the whole lot.

E-mail synchronization through IMAP or Microsoft Exchange mailboxes is just that – synchronization. The whole point is to synchronize things, including deletions. Your iPhone or your Mac may store local copies of your mail, caching mail so you can see some of it even when offline, but they will both always try to synchronize against the e-mail server when possible. (As would any other e-mail client on a Windows PC or Linux P or Android smartphone.)