Add new realm to existing trusted identity provider in SharePoint 2010/2013

Setup: You have a SharePoint farm with an web application configured to use ADFS 2.0 for authentication via Trusted Identity Provider.

Problem: You want to add another web application to this SharePoint farm and configure the authentication via ADFS

Solution:

Open SharePoint PowerShell and issue the following commands:

$ap = Get-SPTrustedIdentityTokenIssuer
$uri = new-object System.Uri("https://<appname>.domain.local")
$ap.ProviderRealms.Add($uri, "urn:sharepoint:<appname>")
$ap.Update()

where appname is the alias of your web application. Then add the Relying Party Trust in ADFS with this new realm->urn:sharepoint:appname.