Hello Klaas, Bandile,
Thank you for posting your question in the Microsoft Q&A Community
From my understanding, you are not able to authenticate to the tenant you were invited to due to the fact that you dont get the 2FA code from your Auth App.
Kindly confirm this statement "On my mobile phone only my home organization appears on my MFA App." Need to be sure you can see the External tenant on the App.
Whichever case, you can contact the Administrator of the External tenant to reset your Authentication method so you are required to re-register for MFA.
Follow the link to get more information about this.
https://learn.microsoft.com/en-us/entra/identity/authentication/howto-mfa-userdevicesettings
Let me know if this helps.
Babafemi