The fighter can also choose her fists as her signature weapon, so if chosen, she deals 1d10 damage with a weapon and her fists.
I also have a house rule that for any class other than the fighter, when a character uses a mundane weapon not listed under their gear lists it counts as awkward and or dangerous (in addition to its other tags). This has no mechanical effect, but has the undeniably awesome fictional tags: It's unwieldy and tough to use & It's easy to get in trouble with it. If you interact with it without proper precautions the GM may freely invoke the consequences of your foolish actions. Once they level up, if they have been stubbornly wielding an uncharacteristic choice, they can choose to add the awkward weapon to their list of 'proficiencies' if they like and remove the tags :)