I have a Exchange 2013 environment that is also running Exchange 2010 coexistence (migrating). What is happening is autodiscover is handing out NTLM for the proxy settings and not basic. However when it is using NTLM we seem to get the password prompt over and over. If I manually changed it to Basic then it works fine, but when autodiscover goes again it changes back to NTLM and prompts that the Administrator made a change and you need to restart Outlook.
I checked Outlook Anywhere and all my servers have Basic set for external users and NTLM set for internal.
I only have a few mailboxes on 2013 and 2010 mailboxes seem not to have a problem.
Here is an output for Outlook Anywhere on all six servers:
Identity : CAS01\Rpc (Default Web Site)
ExchangeVersion : 0.10 (14.0.100.0)
ExternalClientAuthenticationMethod : Basic
InternalClientAuthenticationMethod : Ntlm
IISAuthenticationMethods : {Basic, Ntlm}
Identity : CAS02\Rpc (Default Web Site)
ExchangeVersion : 0.10 (14.0.100.0)
ExternalClientAuthenticationMethod : Basic
InternalClientAuthenticationMethod : Ntlm
IISAuthenticationMethods : {Basic, Ntlm}
Identity : CAS03\Rpc (Default Web Site)
ExchangeVersion : 0.10 (14.0.100.0)
ExternalClientAuthenticationMethod : Basic
InternalClientAuthenticationMethod : Ntlm
IISAuthenticationMethods : {Basic, Ntlm}
Identity : EXCH2K13-01\Rpc (Default Web Site)
ExchangeVersion : 0.20 (15.0.0.0)
ExternalClientAuthenticationMethod : Basic
InternalClientAuthenticationMethod : Ntlm
IISAuthenticationMethods : {Basic, Ntlm, Negotiate}
Identity : EXCH2K13-02\Rpc (Default Web Site)
ExchangeVersion : 0.20 (15.0.0.0)
ExternalClientAuthenticationMethod : Basic
InternalClientAuthenticationMethod : Ntlm
IISAuthenticationMethods : {Basic, Ntlm, Negotiate}
Identity : EXCH2K13-03\Rpc (Default Web Site)
ExchangeVersion : 0.20 (15.0.0.0)
ExternalClientAuthenticationMethod : Basic
InternalClientAuthenticationMethod : Ntlm
IISAuthenticationMethods : {Basic, Ntlm, Negotiate}