With the exception of a handful of beer styles (hazy IPAs and witbiers among them), most modern day consumers and brewers expect beer to be reasonably clear. Haze (temporary or permanent) can be caused by proteins, polyphenols, suspended yeast or even unconverted starch. Aside from aesthetics, removing haze-causing elements in your beer may extend shelf life by reducing potential for oxidation, staling and in some cases bacterial infection.
While time and process are key tools for clarifying beer -- cold crashing and allowing sufficient time for beer to clarify on its own -- there are some other agents that can be added during the boil or fermentation to help in this. Per the comments under ClarityFerm, none of these products are capable of removing gluten from beers made with gluten containing grains in a way that would make them safe for people with medical reasons to avoid gluten. Inclusion here is strictly related to clarity in beers made from naturally gluten free ingredients.
Hot Side Additions
Gluten Free. Vegan. Combination of Irish moss and seaweed-derived carrageenan. Used during latter stages of boil to precipitate proteins.
Gluten Free. Vegan. This is a type of seaweed and one of the main components of Whirlfloc. Used during latter stages of boil to precipitate proteins.
Gluten Free. Vegan. High molecular weight tannic acid that reacts with proteins through adsorption and precipitation. Added in mash, kettle or both.
Cold Side Additions
Normally gluten free but derived from animal collagen hence not vegan. Flavourless gelatin (such as Knox brand) can be rehydrated, heated and added to cold, finished beer prior to packaging. Recommended approach for 5 gallons / 19L is to add 1 to 2 teaspoons of gelatin to 8 oz cool, dechlorinated water. Heat in 10-20 second bursts in microwave to approx 145F/63. For kegged beer, add to an empty and sanitized keg, then pressure transfer previously cold-crashed beer from fermenter to the keg containing the gelatin mix. Pressurize to desired PSI and wait one or two days before dispensing. The first pour will contain the coagulated gelatin and yeast pulled from suspension. The gelatin / water mixture can also be used in carboys prior to racking for bottling, but shaking or rocking the carboy to ensure the gelatin is mixed in may also stir up any yeast or trub at the bottom of the carboy. Make sure you wait for the beer to settle out before racking to another container for bottling.
Derived from carrageenans (Irish Moss seaweed), this gelatin alternative is gluten free and vegan. May also be suitable for clarifying beer.
Normally gluten free but derived from fish (sturgeon bladder) hence not vegan. Comes in liquid or powder form and can be added to the finished beer much like gelatin above (without the need to heat).
ClarityFerm aka Brewer's Clarex
The original intended use of this enzyme was to reduce protein haze in finished beers, and in a beer made from naturally gluten-free grains, it should work perfectly well for that purpose. ClarityFerm / Clarex does not remove gluten from gluten containing grains such as barley, wheat or rye and it's use with those grains does NOT render it safe for people with medical reasons to avoid gluten.
Gluten free and vegan. More popular at the commercial level. Biofine is silicic acid (SiO2) in a water solution.