In versions 11.3 and 12.0, the following causes StreamPlot to fail:

StreamPlot[{
2.1 - 4 u - (8 (u - v) (1 - E^(8 (v - u)^2) + E^(8 (v - u)^2) (2 v - 1)^2 - (2 u - 1)^2))/
(E^(4 (2 u - 1)^2 - 8 (2 u - 1) (2 v - 1) + 4 (2 v - 1)^2) - 1), 
2.1 - 4 v - (8 (v - u) (1 - E^(8 (u - v)^2) + E^(8 (u - v)^2) (2 u - 1)^2 - (2 v - 1)^2))/
(E^(4 (2 u - 1)^2 - 8 (2 u - 1) (2 v - 1) + 4 (2 v - 1)^2) - 1)}
, {u, 0, 1}, {v, 0, 1}]

After some Power::infy and Infinity::indet warnings, the final error is Throw::nocatch -- Uncaught Throw[Null, IntraStepEvaluationException]returned to top level.

In versions 11.2 and before, I get

Yes, the arguments definitely aren't well-defined when u == v (they're automatically generated within another function). However slightly changing any number of things (e.g. replacing 2.1 with 2.2, or changing the limits to {u, 10^-8, 1}, {v, 10^-8, 1}) results in valid output.

Maybe related to this problem?

Any thoughts on a general workaround for this idiosyncratic problem?

In versions 11.3 through 12.1, the following causes StreamPlot to fail:

StreamPlot[{
2.1 - 4 u - (8 (u - v) (1 - E^(8 (v - u)^2) + E^(8 (v - u)^2) (2 v - 1)^2 - (2 u - 1)^2))/
(E^(4 (2 u - 1)^2 - 8 (2 u - 1) (2 v - 1) + 4 (2 v - 1)^2) - 1), 
2.1 - 4 v - (8 (v - u) (1 - E^(8 (u - v)^2) + E^(8 (u - v)^2) (2 u - 1)^2 - (2 v - 1)^2))/
(E^(4 (2 u - 1)^2 - 8 (2 u - 1) (2 v - 1) + 4 (2 v - 1)^2) - 1)}
, {u, 0, 1}, {v, 0, 1}]

After some Power::infy and Infinity::indet warnings, the final error is Throw::nocatch -- Uncaught Throw[Null, IntraStepEvaluationException]returned to top level.

In versions 11.2 and before, I get

Yes, the arguments definitely aren't well-defined when u == v (they're automatically generated within another function). However slightly changing any number of things (e.g. replacing 2.1 with 2.2, or changing the limits to {u, 10^-8, 1}, {v, 10^-8, 1}) results in valid output.

Maybe related to this problem?

Any thoughts on a general workaround for this idiosyncratic problem?