Halliday & Matthiessen (2014: 551-2):
In general, thematic β-clauses serve to set up a local context in the discourse for the α-clause: they re-orient the development (as in the staging of a narrative), often distilling some aspect of what has gone before to provide the point of departure for the dominant clause, thus creating a link to the previous discourse. For example:
||| I remember an example [[[ that happened || when I was probably no more than four years old]]] . ||| My brother and I were playing in a neighbourhood friend’s garage, || and he disappeared for a minute. ||| When our friend came back || he said || that we had to go home, || ‘because my father doesn’t want any niggers in his house.’ ||| We didn’t even know || what the word was. |||
||| The DMK is already annoyed with the BJP government at the Centre || for not favourably considering its demand [[ to recall TN Governor Fathima Bheevi for her swearing in Jayalalitha as CM]] . ||| If the Centre accepts the AIADMK government’s objection || and drops the earlier list, || facilitating the AIADMK government to appoint its choice of judges to the Madras High Court, || then the DMK may voice its opposition to such a move. |||
||| If ifs and ans were pots and pans, || there’d be no need for tinkers. |||
The textual domain of a thematic dependent clause is often a sub-complex rather than just a single clause, and it may even extend beyond the clause complex in which the clause serves.