Must be a dialect thing "you ate already" sounds non-standard without the word "have" to me (I speak a hiberno English dialect)
Q: "Did I eat?" A: "you have ate already" or "you have already ate"
Q "Have I ate?" A "You have already eaten"
I also pronounce "ate" as "et" as opposed to "eight" :-)
You might think that you've been speaking English for a long time, but if you don't recognize the past tense of "eat", you haven't been speaking real English with real English speakers.
What is the past tense of "eat" in this not-really-English language that you've apparently been speaking for a long time?
It's perfect English. "ate" is the past tense of "eat". The sentence is referring to something that happened in the past, so "You ate" is correct. Now, to relay something that happened to me today, my cat, after wolfing down a pouch of catfood, went to the cupboard where I keep her food asking for more, and I said to her, "but you already ate!". Now, I could've used the present perfect and said "you've already eaten", but she spent some time staring out the window at birds, so it wasn't immediate enough to justify the present perfect.
And no, inserting the adverb there doesn't make it ungrammatical.