TLS 1.2 connection problems

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I am trying to establish a TLS 1.2 connection via a machine certificate. This works fine on one machine:

3.590109    ClientIp    ServerIp    TLSv1.2 221 Client Hello
3.617942    ServerIp    ClientIp    TLSv1.2 1514    Server Hello
3.618151    ServerIp    ClientIp    TLSv1.2 1143    Certificate, Server Key Exchange, Certificate Request, Server Hello Done
3.627569    ClientIp    ServerIp    TLSv1.2 3593    Certificate, Client Key Exchange, Certificate Verify, Change Cipher Spec, Encrypted Handshake Message
3.660094    ServerIp    ClientIp    TLSv1.2 105 Change Cipher Spec, Encrypted Handshake Message

Connection established, data flowing. This machine does not use a proxy. The exact same client (a .NET program) on different machine is not so lucky, I can see in Wireshark that the client just does not not seem to answer to the server's certificate request:

10.792079   ClientIp    ProxyIp TLSv1.2 221 Client Hello
10.827628   ServerIp    ClientIp    TLSv1.2 1374    Server Hello
10.827753   ServerIp    ClientIp    TLSv1.2 1374    Certificate [TCP segment of a reassembled PDU]
10.827840   ServerIp    ClientIp    TLSv1.2 103 Server Key Exchange, Certificate Request, Server Hello Done

Then the client reports "The request was aborted: Could not create SSL/TLS secure channel".

The only hint I am seeing is the [TCP segment of a reassembled PDU] message, which I am only (and always) seeing on the machine where the client is not connecting. I read that this just means that Wireshark reassembled packets and that it may not point to any protocol errors, however, maybe it means something since it never appears with the client that can connect?