Communication is one of the strongest sections for real, daily-use apps. SimpleX Chat, Signal and Telegram all publish their iOS source, as do the encrypted email clients Tutanota and Proton Mail. Element and FluffyChat are Matrix clients, Monal speaks XMPP, and Zulip, Mattermost and Rocket.Chat cover team chat. Thunderbird's iOS email client is listed too.
Many of these use GPL or AGPL licenses, which require you to publish changes if you distribute a modified app. The codebases are large and security focused, so they are good for studying end-to-end encryption, push handling and offline sync, but hard to fork lightly. Smaller projects such as Messenger Clone and AppRTC are better starting points if you want to learn real-time chat or WebRTC video calling.