Let's suppose that I am using Tor and I open another page in Firefox at the same time.
- Will it compromise my anonymity?
Actually it can compromise you, if you're using a browser with WebRTC enabled, and the page you're opening has the same tracking code - via WebRTC some leaks are theoretically possible. That's actually why WebRTC is disabled in Tor Browser Bundle
Technically speaking, a Tor site won't know you opened that connection on clearnet and the clearnet site won't be able to tell whether you're using Tor (your ISP can, though). Also you won't have your Tor identity revealed. In theory.
... there's the big problem that you're a human. Humans make errors. If some APT (advanced persistent thread, for example the NSA) or a malicious entity is watching your network it may be possible to trace what you do in Tor and reveal your identity.
There are several ways this could be done:
Now most of these are worst case scenarios, most malicious entities won't be able to fully watch your traffic. Also it's really hard to track your Tor identity.
Coming to a conclusion, it doesn't spoil your identity per-se. But it increases your chances to be revealed by error or wrong behaviour. If you don't do something highly illegal, you're probably safe from this advanced tracking but if you want or need need to stay save in every case, do not do this.