Recommendation for non Touristy Fado place

Unfortunately, pretty much anything you find online and in Lisbon will be touristy. There might be a possibility of a fado concert at a museum - Gulbenkian or Belem Cultural Center - so you might want to check their schedules. If nothing there, check to see if there are venues in the Mouraria area of Lisbon rather than Alfama. Mouraria is still not as touristy and might be a bit better.

Tasca do Chico. You can get by ordering drinks and perhaps an appetizer rather than a full dinner.

I happened by chance upon a very small restaurant called Labios de Mosto at Rua da Oliveira ao Carmo nº51 near Largo de Carmo in Chiado (near the back of Rossio Station). There are frequent Fado performances starting early evening, no cover charge, no pressure at all to buy food or drink, decent food, reasonably priced food and drink. You definitely could just sit at a table with a glass of wine. I was drawn in by the music while passing by. I was especially taken by a woman singer who, I believe, performs there often because a painted portrait of her is on the wall. The man was good too. I'm not sure what counts as non-touristy. On one hand, a waiter was inviting people in off the street which almost put me off and made me think "touristy." But a number of the tables had Portuguese speaking customers.

We asked the same question of our Guesthouse hosts and they recommended Provo on Rua Nova do Carvalho (also known as Pink Street). They were correct. It was a great experience that didn't have the "tourist feel" some of the other places we walked past did. Reservations essential as it was full when the Fado started about 8:30 or 9 (though when we showed up about 30 minutes earlier the inside was empty....lots of people eating/drinking outside, however).