Parse a search string into words and treat words within double quotes as a single term

I know that if I want to parse a string into separate words, the easiest way is.

This is my string:

The Microsoft business analysis software

Code:

SELECT special_term, display_term 
FROM sys.dm_fts_parser ('"The Microsoft business analysis software"', 1033, 0, 0)

Which returns this:

special_term display_term
------------ ------------
Noise Word   the
Exact Match  microsoft
Exact Match  business
Exact Match  analysis
Exact Match  software

But suppose I have a string that has some of the words enclosed in double quotes like this - string:

The Microsoft "business analysis" software

I want to produce the following:

special_term display_term
------------ ------------
Noise Word   the
Exact Match  microsoft
Exact Match  business analysis
Exact Match  software

Is there a reasonable way to do this in SQL Server?

We can "temporarily" replace spaces by some symbol (_) in quoted phrases and replace back after splitting.
Phrase "business analysis" -> business_analysis ->business analysis.

sys.dm_fts_parser output this "words" as Exact Match.

See example

with data as(
  select 1 id
    ,'The Microsoft "business analysis" software And other "helpful programs"' sentence
  union all
  select 2 id
    ,'But suppose I have a string that has "some of the words" enclosed in double quotes like this - string'
)

Recursive replace ' ' by '_' in quoted phrases and remove"

,r as(
  select 0 lvl,id
     ,case when charindex('"',sentence)>0 
         and charindex('"',sentence,charindex('"',sentence)+1)>0
       then left(sentence,charindex('"',sentence)-1)
         +replace(substring(sentence,charindex('"',sentence)+1
            ,charindex('"',sentence,charindex('"',sentence)+1)-charindex('"',sentence)-1)
           ,' ','_')
       +right(sentence,len(sentence)-charindex('"',sentence,charindex('"',sentence)+1))
      else sentence
      end sentence
  from data
  union all
  select lvl+1,id
     ,case when charindex('"',sentence)>0 
         and charindex('"',sentence,charindex('"',sentence)+1)>0
       then substring(sentence,1,charindex('"',sentence)-1)
         +replace(substring(sentence,charindex('"',sentence)+1
            ,charindex('"',sentence,charindex('"',sentence)+1)-charindex('"',sentence)-1)
           ,' ','_')
       +substring(sentence,charindex('"',sentence,charindex('"',sentence)+1)+1,100)
      else sentence
      end sentence
  from r
  where charindex('"',sentence)>0 
)
select id,occurrence,special_term,replace(display_term,'_',' ')display_term 
       --,source_term ,group_id,phrase_id
FROM (select *,row_number()over(partition by id order by lvl desc)rn from r)r
cross apply  sys.dm_fts_parser ('"'+r.sentence+'"',1033,0,0)
where rn=1

Output