Relative to the OP’s concern: “constant current” in pixel chips means that the LED current when on does not vary (much) with changes in the input voltage or LED characteristics (eg: temperature). Some chips provide constant current, some chips provide constant voltage. It’s not a big deal which is used, for most purposes.
This has nothing to do with the “tearing” you are seeing, which is caused by PWM turning the LEDs full ON and and full OFF rapidly, with varying proportions of ON vs OFF to simulate partial brightness (average brightness) thanks to that same visual POV phenomenon (albeit without movement). As people have said, if you use only values of 255 and 0 to create your colors, the tearing goes away. That limits you to 6 colors, plus white and black, tho.
To reduce the tearing even with other brightness values than 255 and 0, you need a faster PWM cycle - it will still have dots, but they will be closer together and usually less visible. The 2811/2812/180x chips have very slow PWM; the 2801 and 8806 and 1829 are much better.
The only exception to this MIGHT be using the 1829 chips, which have an analog current modulation control (varying from 10ma to 41ma in 1 ma steps). This does not give you as many colors as PWM, but it might allow brightness variation while staying fully on (no PWM pulses). It’s something to play with once the library supports them. However, that chip also spec’s a PWM rate of 7 KHz, so it should have virtually unnoticable PWM tearing anyway.
This is why I’m building mine with LPD8806’s and not the slower WS28x series.
Every project has diff requirements, some may favor the denser pitch of 60per Meter at half the price 
I definitely have LPD8806 needs too though for some upcoming schemes.
Chris, there are 8806 based LED strips with 52 pixels/m, which is close. However, nothing is as cheap as the 2812 integrated 5050 RGB LED and 2811 control, so when PWM rate is not important that chip does have advantages.
So the 8806s have a higher frequency PWM driver? Or they are true constant-current (not PWM constant current) drivers?
Is there such thing an as an addressable strip with a true constant current driver (no tearing or PWM in POV images?)
You will drain your battery in a heartbeat that way. 
The whole point of PWM is to save on power. If you can blink the LED fast enough and still get the same amount of light, you will save on power versus leaving it on the whole time. It’s that blink frequency that will help with what you’re referring to as “tearing”. The slower the frequency, the more the dots become obvious. The WS28x series is fairly slow at 2.5KHz. The LPD8806 goes up to 4KHz and is much better for POV installations. And I build my own custom setups as opposed to buying pre-made strips.
Answering your question though: I’m sure there are, however you won’t find an LED strip with using them.
Matt - I tried to answer your question about “constant current” above. “I do not think that word means what you think it means” - at least in the context of LED pixels. Ie: “Constant Current” already has a meaning in this field, and it’s not what you seem to be looking for.
Perhaps you mean something more like “variable current” control of pixel brightness (versus using PWM). (Note that “variable current control of pixel brightness” is NOT an existing term of art in the pixel field like “constant current”, it’s just my ad-hoc description made up on the spot).
So for 50% brightness you would have the pixel select a continuous current which is half the max current (rather than having the max current on 50% of the time and off 50% of the time via PWM).
If so, then the only chipset which claims to do some of that is the 1829, which I do not believe this library supports yet, tho Daniel is planning to do so. And I believe their purpose is more about soft adjustment of pixel hardware, than dynamic control of display. The range is just 10ma to 41ma, and most pixel strings are not going to be able to handle much more than 20ma/color without overheading so the practical range is much less (eg: 10,11,12,…19,20ma which will likely only give about 3-4 discernably different brightness levels).
So basically, the answer is that true dynamic brightness control seems to always be based on PWM with existing chips, and I don’t anticipate that changing much because it’s a good deal more complex and would have side effects (more heat dissipated in the transistors rather than resistors).
But if the PWM is fast enough, it should not matter.
IIRC, the PWM rates are:
2811,2812,1804,1809 - 400+ Hz
2801 - 2500 Hz
8806 - 4000 Hz
1829 - 7000 Hz
Now that’s fascinating! Where’s my on-die LPD8806? ala WS2811. Even cuttable every 3 in 60/led/meter.
I haven’t seen any built-in LPD8806s yet … If you find any, let me know.
Integrated LPD8806s on the LED die would be great; or even the TM1809 for 3 wire systems. I suspect that the WS2812(B) are selling very well, so maybe the other vendors will attempt to get in on that obvious niche.
Not yet tho.
BTW, integrating the chip and LED was the last substantial cost reducer I have been anticipating. From here on, pixels may even start rising in price (as prices for copper and other materials rise faster than chips prices can be driven any further down). Anybody else see any technological steps likely to bring pixel prices signficantly lower than they are today?
Meanwhile. let’s hope some other manufacturers catch up with the 2812B, technology wise.
Last time I placed an order for some LPD8806 ICs, I was told by two separate vendors that they were out of stock and a third one, my ultimate backup, has enough left to fulfill my order. However they indicated that for some reason, the market seems to be running dry on them and they didn’t know (yet) whether it was the factories not being able to keep up or if it’s something else.