While reading the myArmoury review for the Albion Baron it discusses the use of these "Great Swords of War" as intended for two-handed use on the battlefield. Using a weapon with two hands presumably means you're not holding a shield.

In this period (late 13th and early 14th centuries) we have early forms of plate defences but maille is still the primary form of armour for a high-status warrior. Therefore I'd think they'd be rather reluctant to go into battle without a shield.

Any chance these swords were for use in specific scenarios where a shield may not be needed? Seems like a strange and pointless Kamikaze tactic otherwise.

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Many if not most of the men who carried these XIIas and XIIIas into battle did in fact still use shields. There are many images where you can see a heavily armoured soldier with his shield hanging from his neck and welding a large sword with both hands. These swords were probably used single handedly from horseback with a shield. Then if you hand to dismount, you slung your shield over your shoulder and fought with both hands on the sword.

Well if you've already had to dismount, or you've been unhorsed by the enemy, then things have already gone fairly wrong. At this point you're vulnerable no matter whether you have a shield not. You might be faced by enemies on all sides, so a shield slung on your back could protect from unseen blows while you take on what's in front of you with your sword.