English has many grammar and punctuation rules, but there are many exceptions to those rules.
This is what I've always taught in my classes, no exceptions. Now, however, I can't ignore the exception to this rule because it's becoming more common in publications, and my students notice.
If the two independent clauses are short, closely related, and if the comma is omitted, there is no misunderstanding. For example:
He completed the assignment but it was late.
It's not new; it's in the Chicago Manual of Style and the MLA Style Guide, among others, but it's been largely ignored and doesn't come up in most shorter official grammar guides.
The trend is definitely toward minimal punctuation, so I'm getting on board teaching the exception, but only so my students will understand when they see it in print. I will, however, require my students to continue to use the comma in their formal writing assignments so I'm sure they understand the rule, and it's not a purposeful omission. I explain that you can't go wrong with the comma. My students know my mantra: Learn the rules (and demonstrate your understanding) before you break them.