Viral reproductive strategy is never subtle - infect cell, replicate, burst, lather and repeat. In many cases it is the individual immune system response that decides between acute and chronic, symptomatic and asymptomatic etc.
For example, I got exposed to Hepatitis B when I was a child (at least, under 17). Don’t know where, when or how. My immune system jumped on the virus before it got established, and I did not turn yellow and get sick, as most people with Hepatitis B do. But because my immune system did not have to really kick in and fight a raging infection, the HBV was able to hide in my liver for years. I never got sick, but I never got rid of the virus either. And now I rely on antivirals to prevent my liver getting damaged, and will probably do so for the rest of my life.
Hepatitis C is different, only a few people get immediately ill. Most people develop an asymptomatic chronic infection slowly damaging their liver (along with alcohol and drugs, maybe), so that decades after initial infection, they suddenly end up with acute liver disease, and there is very little time to treat them.
Doctors have a suitable array of terms, most of them already mentioned in this thread - chronic (as opposed to acute), subclinical, asymptomatic, occult, latent, suppressed. All perfectly good terms that describe what you want. Just not overarching. But I do like occult. Occult HBV sounds better then chronic HBV (and suppressed occult HBV sounds even better).