This paper presents the experience of verifying client implementations of the TLS cryptographic protocol version 1.3. TLS is a widely used cryptographic protocol today, designed to create secure data transmission channels. The protocol provides the necessary functionality for its tasks: confidentiality of transmitted data, data integrity, and authentication of the parties. In the new version 1.3 of the TLS architecture was significantly redesigned, eliminating a number of shortcomings of previous versions that were identified both during the development of implementations and during their operation. We used a new test suite for verifying client implementations of the TLS 1.3 for compliance with Internet specifications, developed on the basis of the RFC8446, using UniTESK technology and mutation testing methods. To test implementations for compliance with formal specifications, UniTESK technology is used, which provides testing automation tools based on the use of finite state machines. The presented approach has proven effective in several of our projects when testing network protocols, providing detection of various deviations from the specification and other errors. The current work is part of the TLS 1.3 protocol verification project and covers TLS client implementations.