I made an asdf quickstart tutorial a while back that goes through installing languages and Phoenix. I still use it when setting up new machines.

It might help as a quick check to see if there’s something you’re missing.

Erlang 20 with RP.
Elixir 1.5.1.

Remote
mix archive.install https://github.com/phoenixframework/archives/raw/master/phx_new.ez
Are you sure you want to replace it with “https://github.com/phoenixframework/archives/raw/master/phx_new.ez”? [Yn] Y
* creating .asdf/installs/elixir/1.5.1/.mix/archives/phx_new

ls .asdf/installs/elixir/1.5.1/.mix/archives/phx_new/phx_new/ebin/
Elixir.Mix.Tasks.Local.Phoenix.beam Elixir.Phx.New.Generator.beam
Elixir.Mix.Tasks.Local.Phx.beam Elixir.Phx.New.Project.beam
Elixir.Mix.Tasks.Phoenix.New.beam Elixir.Phx.New.Single.beam
Elixir.Mix.Tasks.Phx.New.Ecto.beam Elixir.Phx.New.Umbrella.beam
Elixir.Mix.Tasks.Phx.New.Web.beam Elixir.Phx.New.Web.beam
Elixir.Mix.Tasks.Phx.New.beam phx_new.app
Elixir.Phx.New.Ecto.beam

Local
mix archive.install ~/phx_new.ez

No mix phoenix tasks populated in mix help.

Then create properly filled .tool-versions file in your project root as described in asdfs README:

https://github.com/asdf-vm/asdf#the-tool-versions-file

Maybe you can also set a global version as you require it via asdf global elixir 1.5.1.