The Consecutive Match Bonus is a XP bonus gained for playing matches in a row during one boot of the game. The matches don't have to be in the same lobby, so you can leave a multiplayer match and play campaign and return to multiplayer and the match bonus will stay at what it was last. So, if you play and get a 600XP match bonus, then go play campaign and then switch back to multiplayer, the bonus will still be at 600XP. However, as soon as you go back to dashboard or start a different game, your bonus resets.
Please note that there might be a limit for the time between matches which, if you exceed it, will cause your consecutive bonus to reset. xhunterrrr has reported this, and I will try to verify it in order to get a general value of time. If someone else wants to do testing in addition to mine, that would also be appreciated. It does seem, at least initially, that leaving the game idle has an impact upon this counter, and not switching game modes. That is to say, you should be able to play campaign, horde, etc., without worrying about the bonus resetting, but not playing your console for an extended amount of time will likely reset the bonus.
All you have to do to earn this achievement is play until your bonus hits 1000XP, which will be shown on the post-game XP scoreboard. This takes 11 total games, with the first 100XP awarded after the second match. Once the eleventh game ends and the XP scoreboard appears, the achievement should unlock.
If you're playing private matches and get your match count up to 9 in a row, then play the 10th and 11th match in public, reports are that it will unlock then. The presumed cause is that the bonus just needs to hit 900XP and 1000XP in public matches. Anything before that can be in private matches. Although the consecutive match bonus won't display after a private match, it will still be counted. You can do quick 1v1, 1 round Warzone games against a bot or local player/friend, which don't take much longer than half a minute each, if even that. Credit to https://www.trueachievements.com/DarkxMaterials.htm
for the tip. (He posted it as comment on my solution, then deleted his comment and posted it as own his solution. Give him an up-vote if you feel he deserves it.)
There have also been reports that playing 10 private matches before a single public match will unlock the achievement, but this has not worked for me on multiple accounts, so I recommend sticking with the 9-2 method if you want to make sure you unlock this right away.