DATE, BED, and PRINCESS? With that last as the one word of overlap between Engagement 3 and Sci-fi 6? For the remaining Sci-fi words I like MERCURY, PLOT, and ANTARCTICA.
Since we can't win on this round (we have four guesses maximum), we should go with three words we're reasonably confident about, and end with a word we're less confident about and would be difficult to re-clue, so at worst we rule it out. Which, to me, suggests something like: DATE, BED, MERCURY, and PLOT, in that order.
Edit: Military engagement didn't occur to me. I thought I might be missing something! Yeah, I agree with SOLDIER and MISSILE. PARACHUTE if one of them turns up grey. I'd suggest with DATE, SOLDIER, MERCURY,
PLOT. I think PLOT does become less likely because SCREEN was Red, while MERCURY as a planet should be expected as a Sci-fi guess.
Diamond - ring
Date - setting the date for the wedding. Also going on dates might lead to getting engaged. Weak overlap with sci-fi.
Queen - has Royal Engagements
Soldier - takes part in military engagements. Weak overlap with sci-fi.
Princess - might be expected to get engaged at some point? Good overlap with sci-fi.
We weren't given an unlimited clue, so clearly we're not fully on the right lines with the rest of our Sci-fi answers. I think we should pick Mercury as a safe choice for our +1. We can't risk hitting red again.
After looking at replies:
I'm not a huge fan of Bed.
It's possible Dp101 missed the military context, but whatever we're definitely looking for not just military words, as there would be a much less ambiguous clue available.
I support DIAMOND. It's yet another word I didn't see. That's what comes of browsing these forums so late at night...
It's possible that Dp101 missed either the military (or romantic) connotations but I agree with rho that it's unlikely; there were less ambiguous options available.
Still, I'm more comfortable with the romantic words. So I'm on board with Sunrise's order: DIAMOND/DATE/MERCURY.
We can discuss a fourth word if we get there.