Following the dissolution of the original Covenant, several factions sprung up out of the woodwork to claim the mantle of true 'Covenant'. Whilst the Swords of Sanghelios originally desired to reform the Covenant in their own image, they eventually abandoned their doctrines completely, leaving Jul Mdama as the de-facto leader of the 'Covenant' as he was the most loyal Zealot to the original doctrines and he had the biggest fleet. Although he never referred to his faction as the 'Covenant Remnant', it is regarded as a Remnant force as it is nowhere near as large as the original Covenant and is considered a terrorist sect by the UNSC.
The forces of the Covenant Remnant landed on Requiem and made an alliance with the Didact and the Prometheans. After lengthy battles with both Johnny and the Infinity, and even after the death of the Didact, the Covenant Remnant continued to defend Requiem until they all suddenly decided to evacuate and destroy the planet. Strange. They had all but exhausted their supply of old Covenant warships and so decided to conquer some old Sangheili factory worlds and manufacture their own warships, just as Arbiter and his gang were doing.
Unfortunatly for Jul Mdama, his Promethean forces turned on him in 2558 and in the chaos of their rebellion, Spartan Locke and the rest of Fireteam Osiris were able to infiltrate Jul's camp and kill him, rescuing Doctor Catherine Elizabeth Halsey in the process and effectively cutting the head off what was left of the Covenant. Following this, the Covenant Remnant quickly dissolved, and the final holdout on Sunaion, a city on Sanghelios, was destroyed by the Arbiter's forces during the Battle of Sunaion. Arbiter proclaimed that he would extinguish the Covenant's light forever, and following the rise of Cortana and her Created forces, as well as the Guardians, this appears to be true.
The Remnant or Storm Covenant are not to be confused with Jorgey's Covenant, that also existed on Requiem at the same time. Despite being very similar in terms of armour configuration and weapons, they were two completely different factions that saw very little of each other.