How to shift the phase (185 degrees) of the triangle waveform without a distortion?
Doing a 185° shift (which is an inversion + 5° shift) is hard, because if you actually need your circuit to work 20 Hz to 20 kHz, these 5° can mean delaying by 1/72 of 50 µs or up to 1/72 of 50 ms – you need some circuit to adjust the delay according to your frequency. In the end, you'd build something like a PLL.
In all honesty, get a microcontroller that has a PWM unit that can generate two complementary outputs and has a dead time functionality – it's exactly what you need. That solves the problem you're having before having it.
Or use a microcontroller with two PWM units (that's very standard – 1€ microcontrollers have that) and start one with an offset that gives you the shift you need.
However, assuming your comment:

I do need any solution to end up with phase-shifted waveform. So inverting the input by 180 + fixed time delay would do the trick I guess. So advice about how to delay it would be great.

Since you say you actually don't need the fixed 5° but a fixed delay works, this becomes relatively easy:

invert your signal. That's 180°, for every frequency!
Add a fixed delay.

That fixed delay can be built using a sample-and-hold circuit (better: two in series). You end up with a bucket-brigade device. The cheap and lazy way of solving this is using a microcontroller with built-in ADC and a DAC (or filtered PWM, which is a DAC...) to delay. Again, as you can see, the solution that you generate two triangular waves to begin with becomes more and more tempting.
Note that you'd need to be pretty fast about sampling your input if you need to support a triangle wave of 20 kHz – that has frequency components easily above 200 kHz that you'll need to retain to keep the thing nice and sharp – and thus will need a lot of &quot;memory&quot; (in shapes of charge buckets or actual RAM in a microcontroller). Easier would be to first differentiate the input signal using an opamp to convert the triangle wave to a square wave, and then delay that, and then integrate the result again. As you can see, now we're getting really close to how a triangle wave is originally constructed.