Is it possible to sync a mail account between iPhone 13 and iMac?

If you are using Apple Mail @mac.com, @me.com, @icloud.com it should already be syncing. For 3rd party email accounts you need to use an IMAP account mailbox.

Types of email server connections:

POP3 is designed to access the mailbox, download everything and remove the email from the mailbox. If using more than one device, this means things get out of sync. One method to help with that is to enable the option to keep a copy of those messages on the POP3 mailbox for a set time period. But you will find you will be out of sync regardless. As a second device retrieving the POP3 mail won't know if it's been read, deleted, forwarded, etc. on another device.

IMAP keeps the email on the server and sync's between all clients. You mark an email read on one device and it will be marked read on the other devices and even in the web mail interface. Likewise if you delete an email it deletes it everywhere. iCloud is using IMAP. Gmail simulates IMAP (as they have a proprietary and unique database backed email server).

If you are using the older POP3 type of email account, you should switch to the IMAP configuration instead. POP3 is not designed for sync while IMAP is all about syncing. Most all email providers offer both POP3 and IMAP as connection methods to retrieve your email. You should be able to just add your account again and choose the IMAP option instead of POP3. Once it's working, you can remove the duplicate POP3 account.

Check with your mail server and see if they have the IMAP option. With IMAP mail, changes on one device will be reflected in the other as it just mirrors the centralized account. The other option, and the one you are probably using is POP mail. In this case each device downloads their own copies of emails from the server and there is no communication between devices.

If you do have the option for the IMAP account, you will have to delete the POP mail account on both devices and add the account again as IMAP. This is now the default type to add to the Mail app, but in the past POP was used more often.

This is predicated on the e-mail account is with Apple iCloud

The two links below will offer all the needed information to use iCloud and also to Sync iPhone to any other Qualify device from Apple

Personal note.

Use 4 Different Apple Computers and 1 iPhone.

3 of the 4 Computer are running Ventura macOS 13 and 1 is running High Sierra macOS 10.13.6

The iPhone is running iOS 16.4.1(a) - latest version

All devices sync back and forth without issues

Sign in to iCloud on all your devices - Apple Support (CA)

Apps and features that use iCloud - Apple Support (CA)

Some email providers may require setting up an App password which is different than your normal email password used to access their email via web. Who is your email provider? You mentioned your 'carrier', we may be able to determine if you require an App password or not if we can find the 'carriers' documentation online.

Building on the reply above from @ James Brickley

Google sometimes requires Application Specific Password

This is being a common requirement theses days

https://support.google.com/mail/answer/185833?hl=en-GB

Even Apple does this when Third Party Applications are used to access iCloud e-mail messages

Sign in to apps with your Apple ID using app-specific passwords - Apple Support

Suggest Going into the Web Portal of your phone Carrier E-mail system on your devices.

The Password used to log into that site should be the same Password to using on the computer.

At least that is how it works on my ISP e-mail account.

About midnite to 1 AM Tuesday is my guess.

Perhaps if you and your wife were to go to locale AU Store with iPhone this could sorted out 🤞

My wife is Thai and though I speak Thai reading is off the charts 😔

🇨🇦

You will need the password to set up the IMAP account in Mail. Are you unable to use a web portal of your email client to reset your password?

+1

https://www.au.com/

Sorry but My reading of Japanese is limited

https://www.au.com/mobile/anti-disaster/saigai-dengon/sp-usage/