Caveat: I can't confirm that this works acceptably, as I haven't been able to find any references to anyone who has done it, but by reading the datasheet of the 80C88 it seems it should work there, and it may also work on an original HMOS 8088, but that's less certain as the HMOS design wasn't static (although it could work at relatively slow clock speeds, e.g. 2MHz).
The primary purpose of the 8284 is to provide a clock with 33%/67% duty cycle. The reason for this, per the 8088 datasheet, is "to provide optimized internal timing". That is to say, the chip can run faster because it has an asymmetric duty cycle.
But there is nothing to say that you need the chip to run as fast as it possibly can; if you are happy to run it at a slower rate, there is nothing in the datasheet's timing requirements that suggests that it can't be run with a 50% duty cycle. Just that you have to run it slower in order to do so.
The 8088 (and the modern replacement 80C88) is available in two speed grades: 80[C]88 (max clock speed 5MHz) and 80[C]88-2 (max clock speed 8MHz).
For the 5MHz variant, the requirements for the CLK pin timings are specified as:
Description Min Max Unit
CLK Cycle period 200 500 ns
CLK Low Time 118 ns
CLK High Time 69 ns
CLK Rise Time 10 ns
CLK Fall Time 10 ns
Assuming your clock source has the maximum 10ns rise & fall times, this means that you can meet the requirements with a 50% duty cycle of 118ns. This gives a total clock rate of 1s/(118*2 + 10*2)ns = ~3.9MHz. If you can manage a faster 7ns rise & fall (which should be reasonably easy), that increases to 4MHz, which is easy to produce with readily available crystal oscillators.
For the 8MHz variety, the timings are:
Description Min Max Unit
CLK Cycle period 125 500 ns
CLK Low Time 68 ns
CLK High Time 44 ns
CLK Rise Time 10 ns
CLK Fall Time 10 ns
Again assuming you can manage a 7ns rise/fall time (because it works out neater in this case too), this allows a maximum 6.67MHz (or 6.41MHz with 10ns).
If you can live with this speed for whichever chip you acquire, I see nothing in the datasheet that suggests that using it this way wouldn't work. Then all you need to do is replace the power-on reset functionality with an alternative circuit that meets the 8088's requirements (it needs the RESET line held high for at least 4 cycles at power on in order to initialize correctly) ... a common approach in lower cost computers of the 80s was to use a 555 chip in its monostable configuration (as described in its datasheet) for this purpose.
This suggests that it is relatively easy to do without an 8284. The question is: should you?
If you really want maximum performance, getting one would probably be the easiest way of achieving that. The next best approach would be to use a faster clock (e.g. 15MHz), and either divide it with a counter (counting 0..2 before resetting, with a nor gate to produce a high level signal only when in state 0, for example -- although if you need a 50% duty cycle clock as well, you'll need to run at 30MHz and count to 6, and the logic gets a bit more complex), or run it through a shift register to produce the required output, but that's probably a little bit trickier.
If you are willing to deal with chinese vendors, I see 82C84 chips available on aliexpress.com at sub-$1 prices plus relatively cheap postage. These can be used with either the CMOS or HMOS versions of the processor; they are functionally identical as far as I can see.
If you can live with slow performance, and you'd rather deal with local vendors, 82C84s seem to be hard to acquire. None of the vendors I use regularly stock them, so you're looking at the very least a long wait while they order them in, and you may be looking at a fairly large minimum order. They're also not anywhere near as cheap as the chinese vendors: I see prices of at least $5 here. In this situation, I'd suggest doing without one, and making a reset circuit from a 555, which should be easy to acquire locally and cost a lot less than $5. :)
But there is another way...
As pointed out by @ChrisStratton in comments to @lvd's answer, you can easily use a modern microcontroller to produce the required signals. Obviously this isn't for everyone: in many cases the purpose of projects like this is to find novel ways of using the technology that was available at a particular point in time, which would rule out this approach, but if that's not why you're doing this, then this approach could be the best way of solving this particular problem.
See the comments there for my research on how I came to this design, but it seems if you want to do this, a good way of doing it would be to use a PIC12F1571 chip. This chip is cheap (in the UK, £0.59 in unit quantities, or £0.49 in quantities of 10) and readily available. It can run up to 32MHz, which is fast enough to generate precise clocks for a 5MHz 8088 along with a 50% duty cycle clock for peripherals, or just a CPU clock for running an 8088-2 at very close to its maximum 8MHz frequency.
For 5MHz usage, you'd use it with an external 30MHz crystal, and set up two of its PWM outputs to produce a 2-cycle high, 4-cycle low output (for the CPU clock) and a 3-cycle high, 3-cycle low output (for the peripheral clock).
This uses up 5 of the chip's 8 pins (VCC, GND, and CLK inputs; CCLK and PCLK outputs). There are 3 more available; you can use 2 of them for a reset switch input and RESET signal output and have then entirely duplicated the functionality of the 8284.
For the 8088-2, most things you read would suggest running it at 24MHz with a 1:2 cycle setting on the PWM to produce the CPU clock. Unfortunately this gives results that are slightly out-of-spec -- the clock high phase would be 42ns long, but the datasheet requires 44ns. Now, this would almost certainly work correctly, but to keep within spec you could use a 22.5792MHz crystal to provide an external clock to the MCU. The same 1:2 setting on the PWM would now produce a 44ns/89ns cycle, which is in spec and lets you run the CPU at 7.5264MHz.
(22.5792MHz crystals are apparently quite common: they're 512 * 44100 cycles per second, which makes them useful for running the processors in CD players)
If you really want performance, then you'll need a faster microcontroller still. We're starting to push the limits of what's available without using surface-mount chips, but the PIC18F24K42 is available. It's a DIP-28 chip, so quite large, but maybe you can find other uses for the other pins (turbo switch and output to a 7-segment LED display to show the actual speed, perhaps?). It can run at up to 64MHz; at that speed a 3:5 ratio on the PWM gives a 47ns/78ns clock which gives the maximum 8MHz and is within the CPU's specs, and obviously 4:4 ratio can be used for a peripheral clock. You'd also have plenty of spare PWMs (this chip has 10 of them) which could be used for other useful clocks: a 3-phase 21.333MHz clock could be very handy for implementing a DRAM controller, for example.
