Born in Prague, Réway spent his youth career in Slovakia and with Czech side Sparta Praha, before crossing the Atlantic in 2012.

[citation needed] In 2014, Réway returned to Sparta Praha and made his debut with the men's squad in the Czech Extraliga.

[citation needed] In the following 2017–18 season, Réway made progress in his return to health and was reassigned to begin the year with AHL affiliate, the Laval Rocket.

Having appearing in just 5 games over the first month of the season, Réway re-assessed his option and opted to mutually terminate his contract with the Canadiens in order to return home on 29 October 2017.

[4] Réway returned to playing in the 2018–19 season with hometown club, MHK Dolný Kubín, in the third tier league of Slovakia before agreeing to a one-year contract for the remainder of the year with Swedish outfit, Tingsryds AIF of the HockeyAllsvenskan on 21 November 2018.