How do you navigate your expat identity in Peru?

For 99.99 percent of all Gringo Expats in Latin America:

Yes, you will always be a Gringo.

The locals will spot you and know you are from 'out of town'.

Enjoy your Gringo-ness.  No need to hide it.  Revel in it if you like.

No need to buy indigenous shirts (unless you want to).  No need to wear a Panama hat if you would never wear such in North America or Europe.

Absolutely, learn the language.  But for the 99.99 percent, your accent will always give you away as an out-of-towner.

That's OK.  Being a Gringo has its own status.  You're special. 

And you can still enjoy most of the foods you love (in the cities, anyway) and socialize with whomever you choose (selected from whoever is around).

Back in your home country, some people liked you and probably some didn't.  Whether an individual in a Latino country likes Gringos or not, remember what the Dude said in The Big Lebowski...

"Oh yeah?  Well, it's just like your opinion, man."

  -- cccmedia