Mutual TLS reference architecture

K12140946 is a good place to start. The BIG-IP is a full proxy, so the client mTLS would be terminated there (client = client, BIG-IP = server), and the the BIG-IP would initiate a completely new TLS connection to the backend server (BIG-IP = client, server = server).

As for handling mutual TLS, there are a few options:

• Perform mTLS on the client side to the BIG-IP, and (optionally) perform normal TLS to the backend server. Aside from the C3D option described below, it is otherwise not possible to decrypt mTLS between a client and server. During the mTLS handhshake, the server will request a client cert, the client will abide, and then shortly after send a "signature". This signature is  hash made up of the TLS handshake recorded up to that point, and then encrypted with the client's private key. The server receives this, and assuming it trusts the client's public key (in the certificate), with generate it's own hash, decrypt the client's hash with the client's public key, and then compare the two hashes to verify that the client is who it says it is. It is by virtue of the signature data encrypted with the client's private key that makes it (virtually) impossible to decrypt a mTLS handshake, because no other device would have a copy of the client's private key in order to recreate the signature hash on the server side.
• Perform mTLS on the client side of the BIG-IP and mTLS on the server side with a different cert/key. This server side pair would be manually installed on the BIG-IP. This doesn't do a lot for identifying individual client's, but can satisfy a simple server side mTLS requirement.
• Perform "proxy ssl" with RSA handshakes. This is arguably the worst idea, but it does work. In the era of RSA handshakes, before forward secrecy and TLS 1.3, it was possible for a middle box, with access to the server's private key, to silently record and decrypt TLS handshakes out-of-band. You'd have to force an antiquated RSA handshake on all parties, but in doing so you could indeed passively decrypt the payload between the client and server without breaking the mTLS between them.
• Perform client cert forging with Client Certificate Constrained Delegation (C3D). This is where you perform mTLS on the client side of the BIG-IP, pull the important pieces from that certificate, and use these to issue/sign a new client certificate on the server side. Unlike using a static cert/key, you can dynamically insert the real client cert properties (ie. CN, SAN, policy values, extensions, etc.). C3D will also let you insert arbitrary values as the BIG-IP, sort of like the Kerberos PAC data, where you could insert additional metadata about the client and/or environment that the server could use to further validate the client auth.