They were absolutely real. The bigger ones catered mostly to traveling merchants. They had rooms, halls, and storage areas for goods. They were considered places of "ill repute", and preachers of all faiths regularly cautioned the faithful to stay away from the drinking, gambling, and prostitution that occurred there. Do a search for "fonduks" or "fondacos".

Other than merchant fondacos, "inns" were private residences where you paid to sleep on someone's couch or spare room, basically. It was an accepted part of daily medieval life that you'd rent out spare space in your abode to passing travelers.

We are fortunate to have an amazing work on this subject in English by Olivia Remy Constable, who passed away recently, called Housing the Stranger in the Mediterranean World. The book is readable but dense, and has more information on the subject than you'll ever know what to do with. It's full of interesting insights into daily life in the Middle Ages.

EDIT: After that shining recommendation, I got the title of her book wrong. It's "Mediterranean World", not "medieval world".

You might want to take a look at this thread: https://www.reddit.com/r/AskHistorians/comments/24xlsw/what_common_medieval_fantasy_tropes_have/

It was one of many questions asked there. Both /u/vonadler and /u/Naugrith answered the question but came to slightly different conclusions.

Probably because history classes don't talk much about merchants, the primary users of inns. The other main class of guests in Europe were pilgrims, and on some "pilgrim trails" they were most of the business.

Now, even medievally there were laws regulating professional hospitality. A tavern provided food and drink. An inn provided lodging, and part of that was meals, so also food and drink. A "tavern/inn combo" would be redundant. Next time you're in a hotel or motel, check the sign on the back of the door of your room: you're staying at an inn.

Inns were a distinctly medieval invention. In the worst of the Age of Migration there wasn't a lot of trade and pilgrimage was so difficult you got double points for making one. The source of lodging for strangers was the Church, especially through the monasteries, which built guesthouses when more traffic became usual. The guesthouses were out on the road, keeping the noise of the secular world away. Eventually, these were turned over to laymen to run, and eventually became commercial inns. In England, these can be recognized by signs like the Angel or the Mitre.

Another stream of hospitality came from good Christian lords with bad consciences. For the good of their souls they would build bridges across rivers, causeys thru marshes, and guesthouses for pilgrims. The poor got porridge, bacon and clean straw for free, others paid as they could afford. These would usually have signs like the Bear & Ragged Staff, or whatever badge signified the noble house.

As travel increased, more inns were built for merchants along the trade routes. These might have signs like the Chequerboard (used like an abacus for casting up accounts) or the Sack of Wool.

Come visit the UK. There's a fair chunk of them left.