PNP and NPN H-Bridge - Drive PNP from opposite output
Consider this as the starting scenario: -

Q2 is on and Q4 was off,
Q3 would be activated (due to Q2) and this forces Q1 to be off.

It remains like this for some time.
Then, just imagine Q2 was turned off - ask yourself what will cause Q3 to turn off. With Q2 off, Q3's miller capacitance would remain charged and keep Q3 on for some length of time. How long might that length of time be you might ask. Well, the 10 kΩ resistor feeding Q3 (R8) is not being pulled low by Q2 any more; its kind of floating on the non-base side of Q3.
This creates a long and hard to define CR time with the miller capacitance of Q3; in effect, the base of Q3 is now open circuit because R8 isn't connecting to anything substantial. You might put some number on what that leakage impedance is and, you might choose to look at the 2N2222 data sheet to see if there are any clues. You'd find that in this data sheet the collector leakage current is 10 nA on a 50 volt supply.
That pull-down current will be pulled-up by the leakage current of Q1 (the 2N2907) - it has a collector leakage current of 50 nA. So the net pull-up current is about 40 nA and it is that current that is attempting to discharge the miller capacitance of Q3 and drag Q3's base up towards 30 volts.
So, using this well-known formula for a capacitor: -
$$I = C\dfrac{dv}{dt}$$
We can calculate dv/dt based on the 40 nA and the miller capacitance for Q3 of 8 pF but, that transistor also has a base-emitter capacitance that needs to be discharged of 30 pF. In effect we might be looking at trying to discharge about 40 pF with 40 nA and that means a dv/dt of 1000 volts per second or 1 volt per millisecond.
So, once Q2 turns off it's collector voltage will rise (with a bit of hand-waving) at a rate of 1 volt per millisecond and it will take probably another 29 milliseconds for it to rise enough (from 0 volts to 29 volts) for Q3 to show signs of being deprived of significant base current.
Until this happens you will get a significant shoot-through current should Q4 be activated.

a short dead-time between &quot;on&quot; ~100us

That's a lot shorter than my analysis requires it to be so...
Solution: add base-to-30V-rail resistors on the two PNP transistors - maybe 10k will do but, the lower the better really. 10k implies a discharge current of a few tens of microamp and that's a helluva lot better than 40 nA.