The best examples that come to mind are when ordering food. As examples:
• You speak English and Spanish and are ordering a burrito
• You speak Thai and English and are ordering Tom Yum
I imagine it could depend on numerous things:
• You primary language or ethnicity
• What sort of restaurant
• Who you’re dining with
• Who you’re ordering from
• and probably a lot more…
I say burrito the American-English way because it’s also a word in English. But if I say “arepa” I say it the way it’s pronounced in Spanish.
This trend of pronouncing it the way I first heard extends to dialects, too: Words that I first learned in an Argentine context I tend to pronounce in the Argentine way (eg. the letter elle makes a “zh” or “ĵ” sound) versus the rest of my Spanish which is more (central) Mexican.
I have often irritated or confused acquaintances when using famous quotations or phrases from Latin, which I pronounce in a Classical, rather than Italianate manner (eg. hard Cs and soft Vs).
It’s kind hard to write clearly in print about how we pronounce things, huh?