Most HEMA fencers merely wear more padded versions of the sport fencing style: breeches and a jacket. This style is fine for fencing in a tournament setting or representing fencing as seen in the late Renaissance onward as that was the common fashion (albeit variations in bagginess). However, the fashion of the Middle Ages and Renaissance may influence footwork.
The most common argument for use of historical clothing is in regards to footwear. However, this only applies if one considers the terrain. Turn-shoes will be terrible on a modern gym floor. Also, most people focus on Medieval footwear, whereas in the 17th century for example, high-heel thigh-high boots were the fad and will affect fencing differently
The breeches of the Renaissance are very important to fencing. There are several people out there fencing with plunderhosen. This style, as well as Spanish breeches and Landsknecht pants, affect fencing because there is a possibility of entanglement with an opponent's blade. In a modern sportive context, this would be considered unsafe; in a duel, it may just save one's life, especially if they are made of tough materials. Trunk-hose were a Renaissance fashion which the upper thighs were covered in heavy padding. This would dictate the leg space in a fencing stance.
The last thing to consider wearing when fencing, no matter the period, is a scabbard. Fencing with a scabbard gives 2 insights. First, starting a fight from the draw in the vain of Japanese Iaido. In fact, many treatises teach from the draw such as the first ward of I.33 (now 0.33), Angelo Viggiani, and a section from Fiore Dei Liberi. Second, wearing a scabbard teaches spacial awareness so to not bump into things and to not lodge into the ground. As a bonus, holding down the scabbard is similar to having one's off-hand at the hip as seen in many styles.
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