xelatex
xindy -I latex -o xindy-ipa-sorting.ind -M ipa-alphabet xindy-ipa-sorting.idx
xelatex

documentclass{book}
usepackage{polyglossia}
setmainfont{Times New Roman}
usepackage{makeidx,lipsum}
makeindex
begin{document}
lipsum[1]

index{ʌa}
index{ʌʌ}

index{aa}
index{aʌ}

printindex
end{document}

;; xindy style file for an index with extended alphabet

(define-attributes (("default" "textbf" "textit" "hyperpage")))

;; "see" and "see also"

(define-crossref-class "see")
(markup-crossref-list :class "see" :open "see{" :sep "; " :close "}{}")
(define-crossref-class "seealso")
(markup-crossref-list :class "seealso" :open "seealso{" :sep "; " :close "}{}")

(markup-crossref-layer-list :sep ", ")

(require "base/numeric-sort.xdy")

(define-location-class "arabic-page-numbers" ("arabic-numbers"))
(define-location-class "roman-page-numbers"  ("roman-numbers-lowercase"))
(define-location-class "Roman-page-numbers"  ("roman-numbers-uppercase"))
(define-location-class "alpha-page-numbers"  ("alpha"))
(define-location-class "Alpha-page-numbers"  ("ALPHA"))

(define-location-class-order ("roman-page-numbers"
                  "Roman-page-numbers"
                  "arabic-page-numbers"
                  "alpha-page-numbers"
                  "Alpha-page-numbers"
                  "see"
                  "seealso"))

(require "lang/english/utf8.xdy")

(define-alphabet "latin-ipa"
("ʌ" "a" "b" "d" "dʒ" "e" "ɛ" "f" "ɡ" "ɣ" "h" "i" "j" "k" "l" "m" "n" "o" "p" "q" "r" "s" "ʃ" "t" "tʃ" "u" "ʊ" "w" "x" "z" "ʒ" "ʔ"))
(define-letter-groups
("ʌ" "a" "b" "d" "dʒ" "e" "ɛ" "f" "ɡ" "ɣ" "h" "i" "j" "k" "l" "m" "n" "o" "p" "q" "r" "s" "ʃ" "t" "tʃ" "u" "ʊ" "w" "x" "z" "ʒ" "ʔ"))

(define-letter-group "ʌ" :before "a")
(define-letter-group "dʒ" :after "d" :before "e")
(define-letter-group "ɛ" :after "e" :before "f")
(define-letter-group "ɣ" :after "ɡ" :before "h")
(define-letter-group "ʃ" :after "s" :before "t")
(define-letter-group "tʃ" :after "t" :before "u")
(define-letter-group "ʊ" :after "u" :before "w")
(define-letter-group "ʒ" :after "z" :before "ʔ")
(define-letter-group "ʔ" :after "ʒ" )

;; =======================
;; merge rules
;; =======================

;; LaTeX and TeX conventions

(merge-rule "\LaTeXe *" "LaTeX2e" :eregexp :again)
(merge-rule "\BibTeX *" "BibTeX" :eregexp :again)
(merge-rule "\AmSTeX *" "AmSTeX" :eregexp :again)
(merge-rule "\AmSLaTeX *" "AmSLaTeX" :eregexp :again)
(merge-rule "\XeT *" "XeT" :eregexp :again)

(require "base/tex.xdy")

(markup-locref :open "textbf{" :close "}" :attr "textbf")
(markup-locref :open "textit{" :close "}" :attr "textit")
(markup-locref :open "hyperpage{" :close "}" :attr "hyperpage")

;; =======================
;; sort rules
;; =======================

;; list pali capitals under pali lowercase

(sort-rule "g" "ɡ")

;;(sort-rule "Ā" "ā")
;;(sort-rule "Ṃ" "ṃ")
;;(sort-rule "Ī" "ī")
;;(sort-rule "Ñ" "ñ")

;; list latin capitals under latin lowercase

;;(sort-rule "A" "a")
;;(sort-rule "B" "b")
;;(sort-rule "C" "c")
;;(sort-rule "D" "d")
;;(sort-rule "E" "e")
;;(sort-rule "F" "f")
;;(sort-rule "G" "g")
;;(sort-rule "H" "h")
;;(sort-rule "I" "i")
;;(sort-rule "J" "j")
;;(sort-rule "K" "k")
;;(sort-rule "L" "l")
;;(sort-rule "M" "m")
;;(sort-rule "N" "n")
;;(sort-rule "O" "o")
;;(sort-rule "P" "p")
;;(sort-rule "Q" "q")
;;(sort-rule "R" "r")
;;(sort-rule "S" "s")
;;(sort-rule "T" "t")
;;(sort-rule "U" "u")
;;(sort-rule "V" "v")
;;(sort-rule "W" "w")
;;(sort-rule "X" "x")
;;(sort-rule "Y" "y")
;;(sort-rule "Z" "z")

;; ======================
;; markup rules
;; ======================

(require "base/page-ranges.xdy")

(markup-index :open
"begin{theindex}
  providecommand*lettergroupDefault[1]{}
  providecommand*lettergroup[1]{%
      partextbf{#1}par
      nopagebreak
  }
"
          :close "~n~nend{theindex}~n"
          :tree)

(markup-indexentry :open "~n  item "           :depth 0)
(markup-indexentry :open "~n    subitem "      :depth 1)
(markup-indexentry :open "~n      subsubitem " :depth 2)

(markup-locclass-list :open ", " :sep ", ")
(markup-locref-list   :sep ", ")

;; letter group markup

(markup-letter-group-list :sep "~n~n  indexspace~n")

(markup-letter-group :open-head "~n  lettergroupDefault{" :close-head "}" :group "default")
(markup-letter-group :open-head "~n  lettergroup{" :close-head "}")