This horse is a Marsh Tacky.
Marsh Tacky horses are small compared to other horses, less than five feet tall. Standard is 13 to 15 hands or 52 to 60 inches. They have rather short sturdy legs and a narrow chest.
They come in a lot of different colors; bay, black, chestnut, dun, roan or grullo, like the Spanish horses they descended from. Grullo is tan-gray or mouse-colored, often with some stripes.
They are gentle, with friendly wide-set eyes and a short strong back.
The Marsh Tacky is native to South Carolina. They descended from horses left by Spanish explorers, possibly as early as the 1500s. Perhaps they came from shipwrecks or were left behind by explorers or early settlers. Horses were brought in 1566 when the Spanish tried to settle Santa Elena on Parris Island near Hilton Head. There used to be herds of wild horses on the coast and sea islands.

No doubt these are Marsh Tackies in this drawing of the United States General Hospital on Hilton Head Island during the Civil War.

Most likely these are Marsh Tacky horses.
Union Government buildings to house refugees on Hilton Head Island during the Civil War
Throughout the history of South Carolina, Marsh Tacky horses have been rounded up, tamed and used to ride, as pack horses or to pull carts, wagons or plows. Mounted troops in the American Revolution and the Civil War rode Marsh Tackies. They were farm horses for the Gullah.

During WW2, the Coast Guard trained horses and dogs on Hilton Head Island. Marsh Tackies were calm, brave and confident around the shore, swamps and clamshell beds, making them the ideal companions.
As late as the Second World War, the Beach Patrol rode Marsh Tackies as they watched for Nazi u-boats, spy or troop landings.
“At the Mounted Beach Patrol and Dog Training Center, Hilton Head Island, South Carolina, United States, Coast Guard personnel trained horses and dogs so that they in turn could assist them in the tedious work of patrolling the Southeastern coastline.”
United States Coast Guard
Marsh Tackies used to be the most common horses in South Carolina. There are perhaps only as few as 275 left.
Why were they called Marsh Tacky? Marsh because wild herds lived near the salt marshes of South Carolina. “Tacky” meant inexpensive and common. A Tacky was a common term for an inexpensive horse.
“Tacky: Early 19th century: of unknown origin. Early use was as a noun denoting a horse of little value, later applied to a poor white in some Southern states of the US, hence ‘shabby, cheap, in bad taste’ (mid 19th century).”
The Oxford English Dictionary
The Carolina Marsh Tacky Association holds Marsh Tacky Races on the beach, the Kiawah Cup Beach Race.
You can see Marsh Tackies at the Coastal Discovery Museum. Look in the pasture on the way in.
Coastal Discovery Museum
70 Honey Horn Drive
Hilton Head Island SC
29926
Honey Horn Plantation & Coastal Discovery Museum >
These are near the corner of Squire Pope Road and Gumtree Road, at the entrance to the Spinnaker Bluewater Resort & Marina on the way to Hudson’s restaurant.
How do I know they are Marsh Tacky? I thought so from the look, but as I circled, down the driveway off Gumtree Road there is a sign, just before the No Trespassing sign.
The house is made of tabby. Tabby is a local concrete. Settlers created limed by burning oyster shells. The lime was mixed with water, sand, ash and broken oyster shells to create a very durable concrete.