[racket] typed racket: how to define a binding as both a runtime var and a type?

Using #:omit-define-syntaxes doesn’t work because then when other modules import it it forgets about the type:

def.rkt:
#lang typed/racket
(provide x)
(define-type x 'x #:omit-define-syntaxes)
(define x : x 'x)

ref.rkt:
#lang typed/racket
(require "def.rkt")
x ; works fine, outputs 'x
(define y : x x) ; Type Checker: parse error in type; type name `x' is unbound in: x

Why doesn’t this work, and is there another way to do this?