Just like every autoimmune disease, yes you can have Hashimoto's thyroiditis when you do not have antibodies.
I was one of them. My thyroid exam is very much reminiscent of patients of Hashi yet my TPO antibodies had been negative. If I went to regular physician, I would have been told I have no thyroid disorder.
However, I was in endocrinology fellowship at that time. We have done an Ultrasound on my thyroid gland. This confirmed that I have Hashimoto's thyroiditis. We can tell by looking at the thyroid tissue echogenicity using ultrasound if the gland has autoimmune disease or not.
About 10 years later, my TPO antibodies turned positive.
I have HT, yes but I currently do not need to take thyroid medicine because my thyroid levels are in healthy range and more importantly, I do not have any symptoms of hypothyroidism. I am full of energy and have no fatigue.
If my levels do go down significantly or I develop symptoms, I may need to start taking thyroid hormone but not until then.
I am attaching few US images (dirty looking thyroid is hashi and clean looking thyroid is normal). I don't need to tell you which one is Hashi.