Ask around Tarragona where extranjería is and you may be sent to either of two addresses. Both exist and both deal with foreigners, yet their work differs.
On Plaça d’Orleans sits the Brigada Provincial de Extranjería y Fronteras, which is the Policía Nacional. Its directory entry covers NIE certificates and assignment, EU-citizen matters, TIE cards, return authorisations, students, stay extensions, non-lucrative residence, asylum and travel documents.
At Avenida Estanislau Figueras 59 works the Oficina de Extranjería, part of the Delegación del Gobierno en Cataluña. Its page lists residence authorisations, family reunification, minors, arraigo, long-term residence and renewals among them, plus the EX-19 cards for relatives of Europeans and appeals. That same page sends TIE cards, green certificates, residence and non-residence certificates and the NIE to the police station of the municipality you live in.
In practice a non-EU worker may meet both: one office decides the authorisation, and the police issue the card proving it after the fingerprint sitting. If you are European, or only after a number, you deal with the police alone, at the station of your comarca.
Their timetables differ as well. The Delegación publishes its own, 8:30–17:00 from Monday to Thursday and 8:30–14:00 on Fridays, shorter in summer; the police sources print none, so the time on your confirmation is the one to keep.