Other than completely disabling Outlook Anywhere and requiring everyone to either be on the LAN or connected via VPN to access Outlook mail, is there any way to control access in a similar way as Activesync access can be managed with MDM solutions?

We cannot restrict to only domain joined computers because we have a lot of business partners that use Outlook from laptops joined to their companies' domains.  That is approved because the business partner laptops have similar security policies to ours. What is prohibited is users installing an Outlook client on their privately owned PC and then using it accesses their MAPI account.

Since their unmanaged PC would not be subject to any group policies by us or their employer, they would then be free to then export their mailbox into a PST and do whatever they want with it or simply get their unencrypted home PC stolen with company email data on it.

What options are available for limiting Exchange 2013 MAPI access to specific devices, but not limiting those devices to only devices joined to our own domain?

there must be some 3rd party tool.

In Exchange just give them OWA access only else there is no other option. You want to disable the Outlook Anyhwere for partners and un-managed PC and leave it enabled for your company employees.

OWA is not good enough because OWA doesn't notify you when there is a new email message.  If the partners only had OWA, they would miss lots of time sensitive emails and meeting reminders and we don't want to forward the messages and attachments to their other email address.

don't they have mobile devices and email configured for sensitive email. Everyone use email on their phone and if something has docx or pptx they can go online and use OWA.

So there has to be a some trust and if not then OWA is your only option.

OWA is not good enough because OWA doesn't notify you when there is a new email message.  If the partners only had OWA, they would miss lots of time sensitive emails and meeting reminders and we don't want to forward the messages and attachments to their other email address.

OWA does notify you when a new email arrives. It even makes the message arrival sound.