The only up-armouring which could actually save you from that is ERA. It's lighter than passive steel armour. However it means your vehicles no longer have close infantry protection.
Well you can... just not for long.
Not after the first tile goes off anyway.
What about an early version of Chobham armor/armour? Reactive and ceramic armor were developed/built at about the same time. The ceramics would still be heavy, but not as heavy as a thick steel armor plate.
The US Army experimented with glass armour, utilising both blocks of glass and glassfibre mats on tanks in the 1950s when HEAT was the most common AT round. Worked but was considered too fragile to be able to be used on a battlefield.
The Royal Navy experimented with asphalt armour in the 1940s but found it wasn't effective against AP ammunition.
Chobham's primarily designed to defeat kinetic energy rounds, with a secondary role against chemical energy ones.
I think you have to be clear as to what time period you're talking about. I don't think Chobham was possible before aluminium armour and that doesn't appear until the 1960s when aluminium comes into widespread use in the civilian market and the automatic industry started looking at "crushability" as a safety feature in cars. The combination, plus a bit of lateral thinking created Chobham.
If however, you were talking about the 1950s, an asphalt and glass fibre combination - the glass fibres held in an asphalt matrix might have had possibilities in defeating HEAT rounds, and it would have been durable. Wouldn't work in hot climates though.
Ceramics have real possibilities but don't really start being developed for industrial uses until the 1970s. The Soviets ingeniously used to pour ceramic marbles into a steel matrix for the T-72 turret, so you got a combination of both which made it highly resistant to HEAT and kinetic energy rounds. I wonder, now if someone had thought of that with glass beads, in the 1950s, what effect do you think that would have had on tank design?