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First approach:

\version "2.19.83"

#(define-markup-command (tuplet-bow-up layout props tuplet-number) (markup?)
  (interpret-markup layout props
    (markup #:vspace -1 #:center-column(#:rotate 180 (#:musicglyph "ties.lyric.default")#:vspace -.4 #:line(tuplet-number)))))

#(define-markup-command (tuplet-bow-down layout props tuplet-number) (markup?)
  (interpret-markup layout props
    (markup #:vspace .5 #:center-column(tuplet-number #:vspace -1 #:line(#:musicglyph "ties.lyric.default")))))

tupletBowUp = #(define-scheme-function (parser location tuplet-number) (string?)
     #{\override TupletNumber.text = \markup \tuplet-bow-up #tuplet-number #})

tupletBowDown = #(define-scheme-function (parser location tuplet-number) (string?)
     #{\override TupletNumber.text = \markup \tuplet-bow-down #tuplet-number #})

\score {
  { 
    \clef bass
    \time 3/4
    \tupletBowUp "3" \tuplet 3/2 { c8 b, c } \tupletBowDown "3" \tuplet 3/2 { e8 g e } \tuplet 3/2 { e8 g e } c'4
  }
  
  \layout{}
}

With \tupletBowUp "3" you set the number of all following tuplets to a 3 with a small bow above. With \tupletBowDown "3" you set it to a 3 with a small bow below. You can put other numbers as you like.

Result:

But I am sure, this can also be achieved in a much more sophisticated manner. However, I don’t know the internals of LilyPond that well. For example, it would be great to have the small bow automatically switch above or below the number according to its position.

Edit: Another way would be to change the tuplet bracket like this proposal and one comment suggest. I changed the code from the linked proposal a bit:

printTupletBow = {
  \override TupletBracket #'stencil = #ly:slur::print
  
  \override TupletBracket #'thickness = #1.2
  
  \override TupletBracket #'control-points =
    #(lambda (grob)
      (let* ((x-pos (ly:grob-property grob 'X-positions))
             (pos (ly:grob-property grob 'positions))
             (x-ln (interval-length x-pos))
             (dir (ly:grob-property grob 'direction))
             (height (- (cdr pos) (car pos)))
             (height-corr (* 0.3 dir height))
             (edge-height (ly:grob-property grob 'edge-height '(0.7 . 0.7))))
            
        (list 
          (cons 
            (+ (car x-pos) (* x-ln 1/4)) 
            (+ (car pos) (* 1 dir (car edge-height))))
          (cons 
            (+ (car x-pos) (* x-ln 3/8)) 
            (+ (car pos) (* dir (+ 1.5 height-corr))))
          (cons 
            (+ (car x-pos) (* x-ln 5/8)) 
            (+ (cdr pos) (* dir (- 1.5 height-corr))))
          (cons 
            (+ (car x-pos) (* x-ln 3/4))  
            (+ (cdr pos) (* 1 dir (cdr edge-height)))))))
}

\score {
  {
    \clef bass
    \time 3/4
    \tuplet 3/2 { c8 b, c } \tuplet 3/2 { e8 g e } \tuplet 3/2 { e8 g e } c'4
  }
}

\layout {
  \printTupletBow
}

Result:

One problem with this approach is that the slur will be slanted quite a lot if the notes span bigger intervals. But it might be a nice solution if your piece includes mainly tuplets with beams.