I think I have the same issue:

(use-package ivy
  :custom
  (ivy-re-builders-alist '((swiper . ivy--regex-plus)
                           (t . ivy--regex-fuzzy)))
  :config (ivy-mode))
  ;; :init
  ;; (setq ivy-re-builders-alist '((swiper . ivy--rege-plus)
  ;;                          (t . ivy--regex-fuzzy))))

The only way to make it work is by uncommenting the :init section. Should I move all my config into setq forms in the init section? should I start using ivy-configure (tried to find its documentation at https://oremacs.com/swiper but failed)?

My hydra becomes like this https://imgur.com/a/i5b3HwH for a while. It seems the background and foreground colors are not compatible. Both are very quite light. I don't remember I changed anything directly related to Hydra. Does anybody has some idea about how to solve this? Thanks.

I found it is due to dimmer-mode, together with my new color theme.

Is it possible to define-key ivy-minibuffer-map to open buffer in split windows vertically and horizontally? I've not found any ivy functions that I can use but I'm sure there must be something I'm missing.

auctex changes the default binding of C-M-i to an auctex-specific completion command.

This surely has a good historical reason to be so,

Yes, namely that completion-at-point didn't always exist.

but at this point AUCTeX should probably be providing a simple completion-at-point function instead...

It already does, which is why simply rebinding C-M-i works regardless of Eglot. See (info "(auctex) Completion").

I would be curious to know about your findings.

Okay, but don't hold your breath ;). I probably won't even bother trying to install TexLab until it's easier to do so on Debian, since Digestif already suits my (limited) LaTeX needs.

I can't see that difference reflected in their feature list.

Glad to hear I'm not missing out :). Thanks!

My hydra becomes like this https://imgur.com/a/i5b3HwH for a while. It seems the background and foreground colors are not compatible. Both are very quite light. I don't remember I changed anything directly related to Hydra. Does anybody has some idea about how to solve this? Thanks.

I've yet to compare it to the TexLab [4] server, I guess because Debian's version of rustc is too old to compile it.

I've never done that comparison (I'm the author of Digestif), so I would be curious to know about your findings. TexLab is a much bigger project in terms of lines of code and dependencies, but I can't see that difference reflected in their feature list.

auctex changes the default binding of C-M-i to an auctex-specific completion command.

This surely has a good historical reason to be so, but at this point AUCTeX should probably be providing a simple completion-at-point function instead...

Then when I visit a LaTeX file I just M-x eglot RET and everything works OOTB! :)

Another thing to confirm: does this method also give an incremental drop down completion menu once for all?

It seems to me that the lsp-mode is an indispensable component for this feature, but your above-mentioned configuration don't use it all.

That's because I use the simpler eglot from GNU ELPA (the official Emacs package archive) in place of the larger lsp-mode from MELPA (the popular community archive).

My understanding is that lsp-mode is quite capable and popular, but I always prefer solutions that are as built-in or integrated as possible, and eglot suits all my needs :).

Thank you for sharing valuable experience. I'll try it as soon as possible.

What have you tried?

I give a cursory look at some related packages, including digestif, texlab, lsp-mode, lsp-ivy, and spacemacs, but still can't figure out the workable settings.

It seems to me that the LSP based solution maybe a preferable method over the tricks discussed here.

Sure, whatever works for you.

I sometimes use the Eglot 1 LSP client with the Digestif 3 server.

I've yet to compare it to the TexLab 4 server, I guess because Debian's version of rustc is too old to compile it.

However, TBF, I haven't achieved a feasible configuration based on this method so far.

What have you tried? All I do after installing eglot, auctex, and digestif is:

(with-eval-after-load 'tex
  (define-key TeX-mode-map [remap TeX-complete-symbol] #'completion-at-point))

auctex changes the default binding of C-M-i to an auctex-specific completion command. Since I want eglot LSP-based completion, the snippet above restores the default binding of C-M-i.

Then when I visit a LaTeX file I just M-x eglot RET and everything works OOTB! :)

Have you noticed my latest comments here? It seems to me that the LSP based solution maybe a preferable method over the tricks discussed here.

Then, what's the relationship between company-mode and swiper/ivy?

None, except for both being completion-related. Again, I'm not that familiar with company-mode, but AFAIK it supports different frontends and backends, so I guess Ivy could be used as a frontend and e.g. Counsel functions as backends, or something like that.

That's what company-mode does,

I filed an issue here.

If there aren't any, maybe you can be the first to write one ;).

Maybe I really will do that someday ;-).