No, you really don't. I know the completeness thing is ingrained in nutrition lore and seems to make good sense, but it almost never actually matters.
Try this as an exercise: Suppose you're a 100kg power lifter targeting 1.5 g per kg body weight protein intake (which is at the lowish end of common protein recommendations for athletes, resulting in 150g protein per day in this case). Suppose you don't want to eat anything but beans. Only beans, breakfast lunch and dinner. You make sure you get your 150g of protein per day from beans, which is a lot of beans, but well worth it because you love beans.
By your criteria, your main protein source is horribly incomplete, and I would agree. But now tell me, will you actually end up deficient in any essential amino acids? The answer is, perhaps surprisingly, probably not. Lets briefly look at the numbers:
The limiting EAA in most pulses is methionine. Sometimes tyrosine is low as well, but let's stick with methionine. That means that the only EAA deficiency possible with your bean-only diet is methionine. To get 150g protein you'll need around 600-800g of beans, depending on the type of bean, which yields between roughly 1.2 and 1.8g of methionine, again depending on the type of bean (I'm using USDA data for lentils and pinto beans for this example).
So the question is, is 1.2-1.8 g/day of methionine enough? As far we know, yes, that amount of methionine is at or at least near the amount that your body can use in a day. I've never seen evidence that any amount over that range is of any benefit, any excess over utilization being either oxidized for its caloric value, or excreted. There is evidence that some athletes need more than 1.5 g/kg/day of protein, but this is based on increased oxidation of all dietary amino acids, rather than higher utilization of specific EAAs [1]. So if you need more protein, you can just eat more beans.
In other words, your 600-800g bean-only diet probably gives you complete protein each day. But even if it didn't -- if you somehow turned out methionine-deficient -- then you could easily "complete" your daily protein intake by eating another serving of beans. Or an egg.
Now suppose you're not an athlete, but just a schmuck like me, eating a more typical 0.8 g/kg of protein. Now you're eating only 80g of protein, and thus only around 0.6 to 1.0 g of methionine per day. Now you might end up methionine-deficient, but even here the evidence is thin, and all you'd have to do to correct it is eat an egg or two. Or more beans. (Interestingly, there is great interest in methionine restriction as an anti-aging and anti-cancer strategy, so methionine deficiency is decidedly a rare and speculative condition).
Does this make sense? Do you see how easy it is to consume complete protein each day? This is a specific example, but the arithmetic works out for grain-heavy diets just the same. All you need to do is make sure you're getting the daily amount of protein recommended for your athletic activity level. As long as you do that, you're highly unlikely to be deficient in any EAAs, even on a whack diet. You can live your whole life unaware of protein completeness or amino acid profiles, and your body will never know the difference.
1 - https://onlinelibrary.wiley.com/doi/full/10.1111/nbu.12215, Protein intake for athletes and active adults: Current concepts and controversies