Whether we use stand or stands depends upon what the pronoun that is referring to in the independent clause. This is because that is the subject of the subordinate clause (because it is followed by a verb form) and must have subject agreement with the verb form stand.
The independent clause contains a subject (a phrase) which is defined by a complement (a group of words). Therefore, a phrase = a group of words. They logically describe or define each other. They are one and the same thing.
Since they are one and the same then it might seem logical to conclude that the subordinating pronoun that could refer back to either of them and that we can use either stand or stands to agree with the choice you make.
However, I think it's here that common sense comes into play. We have to choose
the best fit for the whole sentence. In this case I would use a group of words since this phrase adequately fits with the word together in the subordinate clause. The word together describes how something is arranged and implies plurality (you can't put together one thing).
if we use a phrase (= agroup of words) then the sentence just doesn't make sense (what does a phrase stand together with apart from other phrases?).
My choice is therefore to opt for using stand in this sentence to agree with the subject a group of words= that
Your other question
What is the difference between "that" as a subordinator and as a relative pronoun?
****That** is a pronoun that functions as a subordinator (a type of connector)because it connects independent clause with subordinating clauses so that together they contribute to the overall meaning of the sentence.
Furthermore, that also functions as a relative pronoun because it introduces a relative or adjective clause. as part of this function. It also acts as a subordinator as defined above.
In your sentence I believe that is a pronoun which only acts as a subordinator because it introduces an adverbial clause and not an adjective clause. Look at the subordinate clause below in bold
A phrase is a group of words that stand together as part of a sentence
An adverbial answers the question **how? and where?****
How do the words stand? = together
Where do the words stand? = as part of a sentence.
Sources: English Grammar by Betty Azar, Explaining English Grammar by George YUle , CERTESL Grammar 33 Manual University of Saskatchewan.