This is the biggest reason he wasn't allowed "membership" with Russia's Mighty Five that was made up of Rimsky-Korsakov, Borodin, Mussorgsky, Cui, and Balakirev. They would claim that Tchaikovsky's works weren't Russian enough. That he was too heavily influenced by Western European composers. The joke is on them. Tchaikovsky is pretty much a household name and most people know his works even if they don't know the title or his name.
Here's his violin concerto that he composed for his student/lover
You may not know who Rimsky-Koraskov was, but you have almost certainly heard Flight of the Bumblebee and probably would recognize at least the opening of Scheherazade if you heard it and he composed both.