After discovering the truth behind the anchor of Picus, Eliza Cassan, I am confronted by the mercenary Mantis, who has augmentations that match or even defeat mine.
Any damage I do seems quickly repaired, and once she cloaks I can't find her any more.
What tactics should I use to beat her?
Note that Eliza will always warn you before the boss uncloaks to charge you in melee. Try to position yourself around one of the large power cells around the room. When the boss uncloaks to charge you, place yourself between her and the cell, and she'll charge the cell, take damage, and be temporarily disabled (I've found that the easiest way on the hardest difficulty to do this is to jump over her as she charges, but I'm not sure if this is possible without the cybernetic leg prosthesis). Feel free to squeeze off a few head-shots with your weapon of choice while she's incapacitated if you have the electricity-shield upgrade on your dermal armor;otherwise, jump like a bunny on crack if the electricity damages you to avoid taking the full amount of damage. To my knowledge, there is no immediate "safe" place to hide from the electricity damage, but I could be wrong here.
If she still has some health left after charging all four cells, a fairly easy way to get some decent damage in is to stun her with your stun gun or EMP grenades and shoot her a few times with your weapon of choice. Then run. Run like demon monkeys are after you and hide like a scared NPC.
You can also "see" her while she's cloaked if you have the smart vision augmentation; this comes in handy to either plan surprise attacks or get to a safer location, as the boss moves quite slowly while cloaked, even on the hardest difficulty.
It took me about 20 stun gun shots to kill this boss on a non-lethal playthrough.
You haven't mentioned what you have in your inventory or what augments you have chosen.
You should try and listen to her footsteps when she is cloaked. You can easily tell the direction she is approaching from. As soon as she uncloaks, move away from her and try shooting her at the same time. I also read somewhere that you can watch the wet floor for footsteps to find out where she is. Personally, I didn't find this to be of much use.
She usually stops after that initial charge and launches an attack with her augments. If you have an EMP grenade you can disable her for a few seconds and empty your arsenal at her. If you have some kind of fast firing weapon like the heavy rifle or combat rifle or machine pistol, it really helps. You can also move in closer and launch your Typhoon Augment(if you have it) after the EMP.
If you don't have any lethal weapons on you, you can look in the lockers to the sides of the circular room. You will find some weapons and ammo. This is easier if you have cloak augment.
When she is charges and fires at you at the same time just hide, and use your hypostims or painkillers. I think you can find some in the same lockers I mentioned above.
And a bit of advice for the future bosses in the game: even if you are going for the pacifist play-through for the rest of the game, keep lethal weapons in your inventory for the boss fights at least. And EMP and Gas grenades really help in slowing them down.
You can stand there then as soon as she uncloaks and moves towards you. Hit the typhoon and she'll be stunned for a few seconds. Then shoot her in the head with a rapid fire gun. It only took me 2 hits with the typhoon this way to end the fight.
In the recently released Director's Cut version of the game, you can hack a door on either side of the arena. You can find a pocket secretary to give you the code in one of the lockers near one of the doors. Once you're upstairs, look for another pad to hack near the wall. You can either burst through this wall or hack the pad to disable the lasers in the middle of the room. Now hack the computer in the middle of the room and you can turn the turrets against the boss and she should die quite fast.
While doing all this, either use EMP grenades to stun her or avoid her attacks by sprinting right before she attacks you. She will give up after a few attempts.
I found watching the water and chasing her footsteps while she was cloaked helpful for surprise attacks.
I managed to get to this fight but only had a Machine Gun, pistol and 1 health and 2 energy buffs. I worked out how to stun her but didn't have enough ammo and died :(
To stop her, just shoot the power cells which sets off an electrical charge that temporarily disables her, but this also electrocutes you a bit.
I kept dying and dying from the very first power cell electrocution, so I went to a previous save file for enough Praxis and got the EMP Shielding augmentation. Then I was able to survive electrocution and shoot the boss while she was stunned.
After being butchered at least a dozen times, I switched to the heavy rifle even though it usually isn't my weapon of choice. As soon as the cutscene finished, I ran like hell to the right and hacked the door. This gives you access to a ramp leading to the second floor. If I remember correctly it is a level 3 lock, so make sure you have that skill level (you'll need it later this level too). Sprint up the stairs, dropping some EMP or gas mines if you have any. At the top, you should see a code panel on the wall controlling the lasers. You can hack it (level 3) or you can search the locker in the ramp locker for the code.
Once you hack this panel, enough time should have passed that the boss will attack again. Use your heavy rifle and unload the full clip on her. This will stun her for a second, then she will charge again. As soon as she is about to launch her Typhoon, sprint to get out of the way. Unload another clip and she should temporarily retreat.
This will give you enough time to hack the security terminal in the center of the room (also level 3). Switch the turrets to enemy and then leave the computer. Now, all you have to do is wait for her to attack again. Eliza will give you the heads up, the boss will appear, and the turrets will utterly waste her. Personally, I was a little surprised at how hard it was to take her down on my own relative to how quickly the turrets took her down.
You can also use the power cells as outlined in the other answers, but I didn't have the dermal armor or leg upgrades (or skill) to survive such a strategy.
