One thing to keep in mind when accessing Central Admin on SharePoint 2013 (or 2010 for that matter) remotely is that the default URL for Central Admin does not use a FQDN. It uses the server name only. This means that when accessing remotely using the FQDN you will bump into all kind of issues. Especially when Forms Based Authentication is active on a web app in your farm. One of those was the people picker not working no more from the FQDN.
Easy fix! simply add the FQDN to the Alternate Access Mappings for the Central Admin web app and people picker will behave again.
People Picker won't find any people when accessing with the fully qualified domain name instead of the server name.
Existing names are red-underlined although they exist
Error with Event ID 8307 shows up in your Event Logs, mentioning an exception occurred in Forms claim provider when calling SPClaimProvider.FillResolve().