Can the human body significantly build muscle while losing body fat?

The solution of the apparent contradiction may be in the scales of time:

• At any point in time, you can either slim down (catabolic, during excercise) or build muscle (anabolic, during rest after excercise), but
• on a more coarse time scale where excercise and rest phases are "at the same time" both can happen "together".
  See e.g. the paper cited in @SyrenBaran's answer reporting that this does actually happen.

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Roughly speaking,

• catabolism is the part of metabolism (i.e. biochemical pathways) that break down larger molecules into smaller ones, including the part of metabolism that produces free energy.

• anabolism are biochemical pathways that produce larger molecules from smaller ones, using free energy.

By these definitions, β-Oxidation, the process that breaks down lipids (fat) into Acetyl-CoA that can then enter e.g. the citric acid cycle for energy production is catabolic, while producing e.g. muscle protein from amino acids is anabolic.

 catabolic and anabolic pathways are almost always separate. This is required for energetic reasons [...]
Many metabolic reactions are regulated/controlled by the energetic status of the cell.
[...]
ATP-producing [catabolic] pathways are inhibited by high energy charge [that's a way to quantify the energetic status of the cell] whereas ATP-consuming [anabolic] pathways are stimulated.

(Stryer: Biochemie, Spektrum 1991, p. 338, my translation - if someone has access to the English version, feel free to replace)

So, if you look into a muscle cell during excercise, you'll find that mostly catabolic (energy "producing") pathways are "on" in order to rebuild the ATP that is consumed by the physical work, but anabolism (e.g. protein synthesis) is downregulated - that would be a further consumer of energy, which is scarce already. And the need for energy signaled throughout the body, upregulating e.g. β-Oxidation. So during excercise, you slim down but don't build muscle.

On a full body perspective, things get a bit more complicated: for example, you may be breaking down muscle protein into amino acids (catabolic) to have their liver use these amino acids for gluconeogenesis (anabolic) to produce glucose to feed the brain where the glucose enters glycolysis (catabolic). But, if we're looking at cycles of excercise vs. feeding & resting times, we can say that during excercise the metabolism is mostly catabolic, while anabolic pathways are active during feed & rest phases (there are still also catabolic pathways active: they produce the free energy and also substrates for the anabolic processes). See e.g. Rasmussen & Blake: Contractile and Nutritional Regulation of Human Muscle Growth, Exercise and Sport Sciences Reviews: July 2003 - Volume 31 - Issue 3 - p 127-131. They report increased muscle protein synthesis 2 h to up to 2 days after excercise - the review also contains time courses that show muscle protein synthesis ramping up after excercise.

At the first glance, protein metabolism seems to be unrelated to lipid metabolism. But protein is synthetized (anabolic) from amino acids, and these amino acids can also enter catabolic pathways for free energy production - and are then of course no longer available for protein synthesis. These pathways are coupled, and protein synthesis is downregulated when the more pressing need of the body is energy production. You may look at this as avoiding a futile cycle, at least if the energy need is sufficient that the body starts breaking down protein for energy production.

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So the claim "you cannot slim down and build muscle at the same time" is correct in the sense that during excercise*, you can slim down but you don't build muscle. The actual building/growth of muscle happens during feed & rest phases - but only if your energy status is sufficient to not need the amino acids for energy production. I.e. when you don't slim down. The "window" to have the muscle grow and have the fat reserves grow less than what was used up during the excercise does exist, but it is not too wide. Hence the posts that @Fattie links in their answer that emphasize that this will happen only under carefully adjusted conditions.

So as net effect over several excercise and feed & rest cycles (the claim in the question looks at weeks), it is possible to slim down and build muscle.

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* the so-called anabolic excercises have a net muscle growth only over longer excercise plus rest phase cycles. During these excercises, biochemical state is nevertheless catabolic. "Anabolic excercise" IMHO belongs into the same category of terms that calls a lemon an "alkaline food".

It's completely commonplace to increase muscle mass while reducing body fat.

It's trivial to find 1000s of articles and papers on this, random examples https://health.usnews.com/wellness/fitness/articles/can-you-gain-muscle-while-losing-weight , https://www.healthline.com/nutrition/body-recomposition#how-it-works , https://www.verywellfit.com/mistakes-to-avoid-when-building-muscle-and-losing-fat-3498333

The given quote about "anabolism and catabolism", sic, is nonsensical, and indeed way out of context.

1. If you work muscles (and, obviously, get at least enough protein), muscle size will increase.

2. If you cut carbohydrates, and arguably calories, body fat will reduce.

The two are totally unrelated.

(Obviously, there's the ridiculous example where you don't eat ANYTHING, hence zero protein, and of course you cannot increase muscle size, since, muscles are made of protein.)

You only need a relatively small amount of protein to be able to build muscle size; it has no limitation or bearing at all on reducing carbohydrate/calories intake in terms of reducing body fat.

As an extreme example, if you eat a kilogram !!! of chicken breast each day,

https://www.calorieking.com/us/en/foods/f/calories-in-chicken-chicken-breast-without-skin-raw/e4qJu59DSBqeWXUN6cS5iw

You'd be getting an astonishing ~ 1.5 kilograms of protein a week (it is inconceivable your muscle building regime could need more protein than that), and zero carbohydrates and only about 1100 calories a day.

Muscle-building is achieved by muscle-building training (and, sure, you of course have to eat a relatively small amount of protein every day). Body fat reduction is caused by carbohydrate intake (and arguably calorie intake), which has no connection to protein intake.

Yes. That's the normal state for beginners with more than enough body fat and no physical excercise.

Yes, different people get different tips on how to improve, it totally depends on your current physical state.

This even stands true even for older people, e.g. this study comes to the conclusion:

These results support the hypothesis that older persons who consume adequate or moderately high amounts of dietary protein can use RT to improve body composition, oral glucose tolerance, and skeletal muscle aPKC ζ/λ content without a change in body weight.