Hi OP, I work in sexual health 
If I had a penny for every reactive HIV test I’ve repeated and sent to the lab, I wouldn’t need to work as a nurse anymore! Hopefully it will reassure you to know that I’ve not had any come back as true positives, and I have done hundreds of repeats.
Please don’t listen to the posters who are adamant that he is ‘almost certainly positive’ (I’m particularly surprised that the poster who claims to work in GUM would try to scare you like that as reactive does NOT mean false positive, it just means reactive), the home testing kits are a real fucker for reactive results. It does occasionally happen with lab bloods, but it’s less common. We have one patient whose result is reactive every single time we take it so we have to send it off to a special lab for extensive testing each time as he is high risk for a ‘real’ positive so we need to be sure.
Basically, the test works by looking for an indicator of HIV in the blood, almost like a ‘key’ that will unlock the result and show as reactive. If there’s no key, it’s a straight negative. If there’s a key that opens the door, it’s positive. Occasionally, someone will just happen to have a different key that fits in the lock, but doesn’t open the door, which is a reactive. Sometimes the key has disappeared by the time the bloods are repeated, and there is a clear negative. Sometimes the key opens the door in the repeat and it’s a positive. Sorry if that sounds horrendously patronising, it isn’t meant to be! It’s just how I usually explain it to patients in layman’s terms as most people can wrap their heads around it a bit better as it can be really overwhelming.
Please try not to worry about it until you have the repeated result. I know it’s easier said than done!