There is no desktop client yet.
Saying so plainly rather than burying it. Tesseract runs on iPhone and iPad today. A macOS client is being built, Windows comes after it, and a web version is never coming.
macOS — in development
A real client that generates and holds its own keys, not a window onto your phone. No date yet; it ships when the key handling is right.
Windows — planned after macOS
Same protocol, same relay, same end-to-end encryption. Work starts once macOS is out.
Why there will never be a web version.
A web client has to be served to you fresh on every visit, which means whoever serves it decides what code runs against your keys that day. That is not a bug in any particular implementation — it is the shape of the web. A compromised or compelled server can ship one person a build that quietly copies their messages, and there is no practical way for that person to notice.
An installed app can be pinned, inspected and compared against its source. Every Tesseract client holds its own keys on the device it runs on. So: no web version, not now and not later.
A desktop client is not a mirror.
The usual desktop experience is a screen that shows what your phone has already decrypted. That is simpler to build, and it means your phone must be reachable and your keys must effectively live in two places under one lock. The macOS client will have its own identity and its own keys, which is most of why it is taking a while.