Short answer – shoot at night 😉
There’s no way around it – black background in daylight dives is FAR easier with a compact camera, since you can just use 1/2000s to get rid of the background.
With DSLR’s and MIL, the secret is a very strong strobe and being close to the subject.
The 1/160s on the A6*** series is even more limiting than most DSLR’s (which are limited to 1/250s).
It is possible though!
Here’s how you do it:
You set your ISO on lowest, shutter speed on 1/160s and aperture on as low as you need to get a black background.
Then you fire up your strobes all the way to maximum and make sure they light up the subject well enough. This is why for DSLR and Mirrorless it’s recommended to have a powerful strobe, with a GN higher than the entry level 20. The Inon Z-330 or Sea & Sea YS-D2J are perfect choices.
Also, you have to make sure the background is clear behind your subject, otherwise you end up lighting the background as well.
It helps to put the strobes from the side, so they don’t reach the background.
That should give you the desired effect! It works best on macro / close-up shots, but can be done on wide angle as well, depending on lighting conditions.
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