Learn Spanish (Latin America): Me piden que termine el reporte; sin embargo, no tengo tiempo. - They ask me to finish the report; however, I don't have time.

The verb pedir (to ask or request) is a verb of influence. When someone expresses a desire or request for someone else to do something, it triggers the subjunctive mood in the second part of the sentence. termine is the present subjunctive form of terminar.

How do we know who is supposed to finish the report, since termine is the subjunctive form for I, he, she, and you (formal)?

We know from the context of the first half of the sentence. Since they are asking me (Me piden), it naturally follows that they want me to finish it. If it were truly ambiguous, you could clarify by adding the pronoun: que yo termine.

Why doesn't the subjunctive carry over to the end of the sentence? Shouldn't it be no tenga tiempo?

No, because no tengo tiempo is a completely separate clause that states a fact. The influence of Me piden que only applies to the action they are directly requesting (termine). Once we hit the semicolon and sin embargo (however), we are starting a new statement of actual fact, which goes back to the normal indicative mood.

Is sin embargo a fixed phrase, and what exactly does it mean?

Yes, sin embargo is a very common fixed phrase that means 'however' or 'nevertheless'. It is used to contrast two ideas, just like in English. It is a great transitional phrase to start using in your B1 conversations!