No gho or kira required, but cover up at the gate
The gho for men and the kira for women are required dress for Bhutanese citizens in official, religious and school settings. Foreign tourists are exempt. What visitors do face is a modesty code at religious and administrative sites, and the staff at the gate enforce it.
On town streets, in vehicles and on trails, neat casual clothes are fine: t-shirts, hiking pants, jeans without rips. Crop tops, very short shorts and low-cut tops still look out of place in town centers and villages. The real dress code begins at the entrance to a dzong, a temple or a festival ground.
Dzongs and temples: the rules that get enforced
A dzong is both a monastery and the local government office, which is why the guards at the gate care what you wear. Sleeveless shirts, tank tops, vests and low necklines are barred. Shorts and capri pants are out too, so men and women both need full-length trousers or slacks. Skirts and dresses must fall well below the knee, mid-calf to ankle.
Sleeves are where the advice splits. Windhorse Tours says a collarless top such as a crew-neck or henley needs full-length sleeves, while a polo or button-up with a collar can have short ones. Thimphu's Tashichho Dzong is among the strictest, and long sleeves settle the question before it comes up. I'd pack one long-sleeved collared shirt per person and wear it to every dzong.
Hats, caps and sunglasses come off in courtyards, temples and shrine rooms. Shoes come off before any temple interior or inner chapel, and you should wear socks. Flip-flops and open sandals stay at the hotel, since enclosed shoes are expected in dzongs.
For US readers, the smart-casual that dzongs expect is closer to business-casual: a button-down or polo, chinos or slacks, or a modest maxi dress. Jeans are the shaky choice, because some operators encourage chinos or tailored trousers to match what locals wear. Skip the debate and bring chinos.
Your guide acts as a checkpoint too. Some guides look over outfits before leaving the hotel and won't take guests to a dzong in a crew-neck t-shirt or shorts, so dress for the dzong before you leave and you won't be changing in the parking lot.
Tiger's Nest and temple-hopping days
Carry a packable temple layer in your daypack on the Tiger's Nest hike: a collared overshirt, lightweight long pants if you hike in shorts or tights, and a pair of socks. Hike in athletic layers, then change before the inner chapels. Kiyul Odyssey's packing guide says hikers in convertible pants who hadn't zipped the legs back on were held outside until they were fully covered.
On valley sightseeing days, slip-on shoes or trail shoes with elastic laces save time at every temple doorway. Stiff lace-up boots are a nuisance there. From October to April the stone and wood floors are unheated, so thick wool socks are worth packing.
Festivals and trying on national dress
Tshechu festivals are held in dzong courtyards, and locals attend in their best silks and jewelry. Leave the trekking clothes in the bag and dress up: a blazer, dress shirt, smart slacks or a long dress.
If you want to wear a gho or half-kira, hotels and guides can arrange rentals. Skip the sash. The kabney (men) and rachu (women) signal rank, so foreigners shouldn't wear them.