Does the sun have a shadow?

Technically, like any non-transparent object, the Sun would cast a shadow on another object. However, since the Sun emits light itself, it would only result in a true shadow if there were a stronger level of illumination on that object coming from a source behind the Sun.

So theoretically, it does cast a shadow, practically it would only do so if a brighter star (or other source of light) were to appear or move near it.

(Can't really quote a reference on this one - it's essentially logic and the definition of "shadow")

Shadows are the effect of an interruption of a beam of light. It means you can no longer see the source of the light. Move out of the shadow and you can see the light source. Even something coming into the path of a laser beam creates a shadow behind it. Distance from the light source is irrelevant. Therefore, the sun creates shadows associated with every star shining at it. The sun (Sol) occults the light from Alpha Centauri and so a viewer on the far side of the sun from A Centauri and could nor see it, would be in the shadow cast by the sun when it interrupted A Centauri's light. So, depending on how far behind the sun you are, you are in the shadow of everything shining light on the other side of it. A galaxy billions and billions of light years distant will cast a shadow when the sun comes between (occults) the viewer and the light source. The sun is casting as many shadows as there are light sources, no matter how dim or distant, constantly. ................. https://en.wikipedia.org/wiki/Shadow ................. https://en.wikipedia.org/wiki/Occultation
Sun A emits 20 megalumens; Sun B emits 20 megalumens. Together shining on object 1 they cast 40 megalumens. If Sun A occults Sun B from the perspective of Object 1, Object 1 only has 20 megalumens shining on it from Sun A because it is in the shadow of Sun A and the radiance of Sun B is blocked. It is still lit but that isn't the Question. The Question was does the Sun cast a shadow and technically by the mentioned logic and arithmetic it does cast a shadow by occulting Sun B.