I would suggest to the OP to read Koopmans, L. H. "Teaching singular distributions to undergraduates." The American Statistician 37, no. 4a (1983): 313-316.

My general impression is that "CDF without densities" arise in a way that can be "real-world intuitive" in multivariate settings rather than univariate. As he writes (p. 314)

Finally, whereas singular distributions are difficult (if not impossible) to visualize in one dimension, being distributions concentrated on uncountable zero- dimensional sets (so to speak), in two dimensions they are, or can be constructed to be, concentrated on rather familiar one-dimensional figures such as lines and circles.

He also provides a real-world example indeed (Example 1, p. 314).

Elaborating in a naive way on the bivariate CDF with which he starts, he considers

$$F(x,y) = \frac{x+y}{2},\;\;\;\; 0\leq x\leq 1,\;\; 0\leq y\leq 1.$$

Don't be misled by the notation that "points" towards "continuous distributions". After all this is exactly the issue here.

One can verify that the cross-partial of $F(x,y)$ is zero, $$\frac{\partial^2 F(x,y)}{\partial x \partial y} =0.$$

Strictly speaking the mathematical operation of computing the derivative appears legal, given the information we have, but by giving us the constant zero-function, it leaves us scratching our heads (Koopmans writes "At this point in an exam, it is not unreasonable to expect the normal undergraduate to panic.")

An indication that something peculiar may be going on here is by deriving the marginal CDF, say, for $X$, $$\Pr (X\leq x) = G(x) = \lim_{y\to 1}F(x,y) = \frac{x}{2} + \frac 12.$$ But his means that $\Pr (X=0) = \frac 12$. This implies that $X$ is a random variable of "mixed" type (not mixture), and that its marginal CDF of $X$ has one continuous and a non-zero discrete component: $$F(x) = \begin{cases} 1/2\;\;\;\; x=0 \\ x/2\;\;\;\; 0 < x \leq 1.\end{cases}$$

Koopmans shows that such a mixed distribution leads to a non-zero "singular" component, in the Lebesgue decomposition of a joint density in three parts, one continuous, one discrete and one singular.

I would suggest to the OP to read Koopmans, L. H. "Teaching singular distributions to undergraduates." The American Statistician 37, no. 4a (1983): 313-316.

My general impression is that "CDF without densities" arise in a way that can be "real-world intuitive" in multivariate settings rather than univariate. As he writes (p. 314)

Finally, whereas singular distributions are difficult (if not impossible) to visualize in one dimension, being distributions concentrated on uncountable zero- dimensional sets (so to speak), in two dimensions they are, or can be constructed to be, concentrated on rather familiar one-dimensional figures such as lines and circles.

He also provides a real-world example indeed (Example 1, p. 314).

Elaborating in a naive way on the bivariate CDF with which he starts, he considers

$$F(x,y) = \frac{x+y}{2},\;\;\;\; 0\leq x\leq 1,\;\; 0\leq y\leq 1.$$

Don't be misled by the notation that "points" towards "continuous distributions". After all this is exactly the issue here.

One can verify that the cross-partial of $F(x,y)$ is zero, $$\frac{\partial^2 F(x,y)}{\partial x \partial y} =0.$$

Strictly speaking the mathematical operation of computing the derivative appears legal, given the information we have, but by giving us the constant zero-function, it leaves us scratching our heads (Koopmans writes "At this point in an exam, it is not unreasonable to expect the normal undergraduate to panic.")

An indication that something peculiar may be going on here is by deriving the marginal CDF, say, for $X$, $$\Pr (X\leq x) = G(x) = \lim_{y\to 1}F(x,y) = \frac{x}{2} + \frac 12.$$ But this means that $\Pr (X=0) = \frac 12$, implying that $X$ is a random variable of "mixed" type (not mixture), and that its marginal CDF $G(x)$ has one continuous and a non-zero discrete component: $$G(x) = \begin{cases} 1/2\qquad\;\;\;\;\;\;\; x=0 \\ 1/2 + x/2 \;\;\;\; 0 < x \leq 1.\end{cases}$$

Koopmans shows that such a mixed distribution leads to a non-zero "singular" component, in the Lebesgue decomposition of a joint density in three parts, one continuous, one discrete and one singular.